I.Division of a Segment in a Given Ratio

If two points \(A(x_1,y_1)\) and \(B(x_2,y_2)\) are given, the point \(C(x,y)\) on the same line dividing the segment in the ratio \(\lambda\) has coordinates:

II.General Equation of a Line

Every equation of the first degree in \(x\) and \(y\), i.e. an equation of the form \(Ax+By+C=0\) where \(A,B,C\) are constants, defines a certain line in the plane. This is called the general equation of a line. Particular cases:

• \(C=0,\ A\neq0,\ B\neq0\): the line \(Ax+By=0\) passes through the origin.
• \(A=0,\ B\neq0,\ C\neq0\): the line \(By+C=0\) is parallel to the \(Ox\) axis.
• \(B=0,\ A\neq0,\ C\neq0\): the line \(Ax+C=0\) is parallel to the \(Oy\) axis.
• \(B=C=0,\ A\neq0\): the line \(Ax=0\) coincides with the \(Oy\) axis.
• \(A=C=0,\ B\neq0\): the line \(By=0\) coincides with the \(Ox\) axis.

III (a).Equation of a Line with Slope

The equation has the form \(y=kx+b\), where \(b=-\dfrac{C}{B}\) and \(k=-\dfrac{A}{B}=\tan\alpha\). Here \(A,\,B,\,C\) are the coefficients of the general equation of the line, and \(\alpha\) is the angle formed by the line with the positive direction of the \(Ox\) axis. The free term \(b\) equals the ordinate of the point where the line meets the \(Oy\) axis.

III (b).Line with Slope \(k\) Through a Point \(M(x_1,y_1)\)

III (c).Line Through Two Points \(M_1(x_1,y_1)\) and \(M_2(x_2,y_2)\)

The slope is \(k=\dfrac{y_2-y_1}{x_2-x_1}\). If \(x_1=x_2\) the line is vertical, \(x=x_1\); if \(y_1=y_2\) it is horizontal, \(y=y_1\).

IV.Equation of a Line in Intercept Form

An equation of the form

where \(a=-\dfrac{C}{A}\) and \(b=-\dfrac{C}{B}\) (with \(A,\,B,\,C\) the coefficients of the general equation of the line), is called the equation of a line in intercept form. Here \(a\) is the abscissa of the point where the line meets the \(Ox\) axis, and \(b\) is the ordinate of the point where it meets the \(Oy\) axis.

V.Normal Equation of a Line

An equation of the form

is called the normal equation of a line. Here \(p\) is the length of the perpendicular dropped from the origin onto the line, and \(\varphi\) is the angle formed by this perpendicular with the positive direction of the \(Ox\) axis.

To obtain the normal equation from the general equation \(Ax+By+C=0\), both sides are multiplied by the normalising factor

The sign before the radical is chosen so that the condition \(\mu C<0\) holds.

VI (a).Angle Between Lines — Slope Form

The acute angle \(\varphi\) between the lines \(y=k_1x+b_1\) and \(y=k_2x+b_2\) is found by the formula

The lines are parallel when \(k_2=k_1\), and perpendicular when \(k_1=-\dfrac{1}{k_2}\).

VI (b).Angle Between Lines — General Form

If the lines are given by the equations \(A_1x+B_1y+C_1=0\) and \(A_2x+B_2y+C_2=0\), then the angle \(\varphi\) between them is

The lines are parallel when \(A_1B_2-A_2B_1=0\) (equivalently \(A_1/A_2=B_1/B_2\)), and perpendicular when \(A_1A_2+B_1B_2=0\).

VII.Intersection of Lines

The coordinates of the intersection point of \(A_1x+B_1y+C_1=0\) and \(A_2x+B_2y+C_2=0\) are found by solving the system

which gives

Comparing the ratios of coefficients:

• \(\dfrac{A_1}{A_2}\neq\dfrac{B_1}{B_2}\): the lines intersect at one point.
• \(\dfrac{A_1}{A_2}=\dfrac{B_1}{B_2}\neq\dfrac{C_1}{C_2}\): the lines are parallel.
• \(\dfrac{A_1}{A_2}=\dfrac{B_1}{B_2}=\dfrac{C_1}{C_2}\): the lines coincide.

VIII (a).Deviation / Distance of a Point from a Line

The deviation of a point \(M(x_0,y_0)\) from the line \(Ax+By+C=0\) is found by the formula

where the sign before the radical is chosen as in the normal equation (§V). The distance \(d\) from the point \(M\) to the line then equals \(d=|\delta|\).

VIII (b).Bisectors of the Angles Between Lines

The choice of sign is made according to the figure: a point on a bisector is equidistant from both lines, \(|d_1|=|d_2|\).

IX.Pencil of Lines

If two intersecting lines are given by the equations \(A_1x+B_1y+C_1=0\) and \(A_2x+B_2y+C_2=0\), then the equation

defines a line passing through the point of intersection of the given lines. Here \(\lambda\) is a numerical parameter. The set of all values of \(\lambda\) defines the pencil of lines, whose centre is the point of intersection of the given lines.

I (a).General Equation of a Plane

Every equation of the first degree with respect to the coordinates of a point in space, \(Ax+By+Cz+D=0\), defines a plane; conversely, every plane can be described by an equation of the first degree. Here \(A,\,B,\,C\) are the coordinates of the normal vector:

I (b).Condition for a Point to Lie on a Plane

A point \(M_0(x_0,y_0,z_0)\) lies on the plane \(A_1x+B_1y+C_1z+D_1=0\) precisely when its coordinates satisfy the equation:

II.Normal Equation of a Plane

The normal equation of a plane has the form

where \(\cos\alpha,\,\cos\beta,\,\cos\gamma\) are the direction cosines of the normal and \(p\) is the distance from the origin to the plane:

This equation is obtained from the general equation \(Ax+By+Cz+D=0\) by multiplying both sides by the normalising factor \(\mu=\pm\dfrac{1}{\sqrt{A^2+B^2+C^2}}\). The sign of the normalising factor is taken opposite to the sign of the free term \(D\) of the general equation.

III (a).Equation of a Plane in Intercept Form

The intercept form of the equation of a plane is

where \(a=-\dfrac{D}{A}\), \(b=-\dfrac{D}{B}\), \(c=-\dfrac{D}{C}\). The numbers \(a,\,b,\,c\) are respectively the abscissa, ordinate, and applicate (the \(x\)-, \(y\)- and \(z\)-intercepts) of the points where the plane meets the coordinate axes.

III (b).Plane Through a Point, Perpendicular to a Vector

The equation of the plane passing through the point \(M_0(x_0,y_0,z_0)\) and perpendicular to the vector \(\overline{n}=A\overline{i}+B\overline{j}+C\overline{k}\) has the form

or, in vector form, \(\bigl(\overline{n},\,\overline{M_0M}\bigr)=0\).

View in 3D ▸

III (c).Plane Through Three Points

The equation of the plane passing through three given points \(M_1(x_1,y_1,z_1)\), \(M_2(x_2,y_2,z_2)\), \(M_3(x_3,y_3,z_3)\) has the form

or, in vector form, as the vanishing of the scalar triple product \(\bigl(\overline{M_1M_2},\;\overline{M_1M_3},\;\overline{M_1M}\bigr)=0\).

III (d).Pencil of Planes

For an arbitrary value of the parameter \(\lambda\), the equation

defines a plane passing through the line of intersection of the planes

III (e).Condition for a Point to Lie on a Plane of the Pencil

The condition for a point \(M_0(x_0,y_0,z_0)\) to lie on a plane belonging to the pencil has the form

From this we find the corresponding value \(\lambda_{M_0}\):

III (f).Plane Through a Point and a Line

The equation of the plane passing through the point \(M_0(x_0,y_0,z_0)\) and the line defined as the intersection of the two planes \(A_1x+B_1y+C_1z+D_1=0\) and \(A_2x+B_2y+C_2z+D_2=0\) has the form

IV.Angle Between Two Planes

The angle \(\varphi\) between the planes \(A_1x+B_1y+C_1z+D_1=0\) and \(A_2x+B_2y+C_2z+D_2=0\) is determined by the formula

The planes are parallel when

perpendicular when

and intersecting when

View in 3D ▸

V.Deviation of a Point from a Plane

The deviation \(\delta\) of a point \(M_0(x_0,y_0,z_0)\) from the plane \(Ax+By+Cz+D=0\) is found by the formula

where the sign before the radical is taken opposite to the sign of the free term \(D\). The distance from the point \(M_0\) to the plane equals \(d=|\delta|\).

VI (a).Line Through Two Points

The equation of the straight line passing through two points \(M_1(x_1,y_1,z_1)\) and \(M_2(x_2,y_2,z_2)\) has the form

or, in vector form,

where \(\overline{M_1M}=(x-x_1,\,y-y_1,\,z-z_1)\) and \(\overline{M_1M_2}=(x_2-x_1,\,y_2-y_1,\,z_2-z_1)\).

VI (b).Canonical Equation of a Line

The equation of the straight line passing through the point \(M_1(x_1,y_1,z_1)\) parallel to the vector \(\overline{S}=l\overline{i}+m\overline{j}+n\overline{k}\) has the form

This is the canonical equation of the line; the vector \(\overline{S}\) is called the direction vector of the line \(L\).

VI (c).Parametric Equation of a Line

The parametric equation of a straight line is

It is obtained from the canonical equation by introducing the parameter \(t\): setting \(\dfrac{x-x_1}{l}=\dfrac{y-y_1}{m}=\dfrac{z-z_1}{n}=t\) and solving each ratio for the corresponding coordinate.

VI (d).Line as the Intersection of Two Planes

A straight line in space can be defined by the equations of two planes:

This is the general equation of a straight line. To bring it to canonical form:

1) Find a point \(M\) on the line by fixing one of the coordinates (say \(z=z_0\)) and solving the system for the remaining two:

2) Find the direction vector \(\overline{S}\), parallel to the line (\(\overline{S}\perp\overline{n}_1\) and \(\overline{S}\perp\overline{n}_2\)), as the cross product

Hence \(\dfrac{x-x_0}{l}=\dfrac{y-y_0}{m}=\dfrac{z-z_0}{n}\) — the canonical form of the line.

VI (e).From Canonical Form to Two Planes

If a line is given by the canonical equation \(\dfrac{x-x_0}{l}=\dfrac{y-y_0}{m}=\dfrac{z-z_0}{n}\), then the pair of equations

defines the same line as the intersection of two planes.

VI (f).Angle Between Two Lines

The angle between two lines in space, given by their canonical equations \(\dfrac{x-x_1}{l_1}=\dfrac{y-y_1}{m_1}=\dfrac{z-z_1}{n_1}\) and \(\dfrac{x-x_2}{l_2}=\dfrac{y-y_2}{m_2}=\dfrac{z-z_2}{n_2}\), is determined by the formula

where \(\overline{S}_1=(l_1,m_1,n_1)\) and \(\overline{S}_2=(l_2,m_2,n_2)\). The lines are parallel when

and perpendicular when

VI (g).Condition for Two Lines to be Coplanar

Two lines given by their canonical equations \(\dfrac{x-x_1}{l_1}=\dfrac{y-y_1}{m_1}=\dfrac{z-z_1}{n_1}\) and \(\dfrac{x-x_2}{l_2}=\dfrac{y-y_2}{m_2}=\dfrac{z-z_2}{n_2}\) are coplanar if and only if

If the numbers \(l_1,m_1,n_1\) are not proportional to \(l_2,m_2,n_2\), this coplanarity condition is the necessary and sufficient condition for the two lines to intersect.

VII.Angle Between a Line and a Plane

The angle between a line \(\dfrac{x-x_1}{l}=\dfrac{y-y_1}{m}=\dfrac{z-z_1}{n}\) and a plane \(Ax+By+Cz+D=0\) is determined by the formula

where \(\overline{S}=\{l,m,n\}\) and \(\overline{n}=\{A,B,C\}\). The line is parallel to the plane when

and perpendicular to the plane when

View in 3D ▸

VIII.Point of Intersection of a Line and a Plane

To find the point of intersection of a line \(\dfrac{x-x_0}{l}=\dfrac{y-y_0}{m}=\dfrac{z-z_0}{n}\) with a plane \(Ax+By+Cz+D=0\), write the line in parametric form

and substitute into the equation of the plane. From it we determine the parameter \(t\):

Substituting this value back into the parametric equations gives the coordinates \(x,\,y,\,z\) of the intersection point.

• If \(Al+Bm+Cn\neq0\), the line intersects the plane.
• If \(Al+Bm+Cn=0\) and \(Ax_0+By_0+Cz_0+D\neq0\), the line is parallel to the plane.
• If \(Al+Bm+Cn=0\) and \(Ax_0+By_0+Cz_0+D=0\), the line lies in the plane.

I.1.Coordinates of a Vector

If a vector \(\overline{a}\) has Cartesian rectangular coordinates \(x,\,y,\,z\), then

If a vector \(\overline{AB}\) is given by its initial point \(A(x_1,y_1,z_1)\) and terminal point \(B(x_2,y_2,z_2)\), then the coordinates of this vector \(\overline{AB}=\{x,y,z\}\) equal the differences of the like coordinates of the end and the start:

The zero vector is a vector whose start and end coincide (it has an arbitrary direction).

I.2.Sum and Difference of Vectors

Sum. The sum of two vectors is built either by the triangle rule (place them head to tail) or by the parallelogram rule (draw them from a common origin).

Difference. The difference \(\overline{c}=\overline{a}-\overline{b}\) is the other diagonal of the parallelogram built on \(\overline{a}\) and \(\overline{b}\).

If \(\overline{a}=\{x_1,y_1,z_1\}\) and \(\overline{b}=\{x_2,y_2,z_2\}\), then

If \(\overline{a}=\{x_1,y_1,z_1\}\) and \(\lambda=\text{const}\), then

I.3.Length of a Vector

If \(\overline{a}=\{x,y,z\}\), the length of the vector in an orthonormal basis is

Since

we have \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\). The direction cosines of a vector are the coordinates of its ort (unit vector):

I.4.Scalar Product of Vectors

Definition. The scalar product of two vectors is the number equal to the product of their lengths and the cosine of the angle between them:

Properties of the scalar product:

1. \(\overline{a}\cdot\overline{b}=\overline{b}\cdot\overline{a}\) — commutativity;
2. \((\lambda\overline{a})\cdot\overline{b}=\lambda(\overline{a}\,\overline{b})\) — associativity;
3. \(\overline{a}(\overline{b}+\overline{c})=\overline{a}\,\overline{b}+\overline{a}\,\overline{c}\) — distributivity;
4. \(\overline{a}\cdot\overline{b}=0\Leftrightarrow\overline{a}\perp\overline{b}\), or \(\overline{a}=0\), or \(\overline{b}=0\);
5. \(\overline{a}\cdot\overline{a}=|\overline{a}|^2\Rightarrow|\overline{a}|=\sqrt{\overline{a}\cdot\overline{a}}\).

If \(\overline{a}=\{x_1,y_1,z_1\}\) and \(\overline{b}=\{x_2,y_2,z_2\}\), then

I.4 (a).Length of \(\overline{a}=\alpha\overline{m}+\beta\overline{n}\)

If \(\overline{a}=\alpha\overline{m}+\beta\overline{n}\), where \(|\overline{m}|,\,|\overline{n}|\) and the angle \(\widehat{\overline{m},\overline{n}}\) between the vectors are known and \(\alpha,\beta=\text{const}\), then

I.4 (b).Scalar Product of Two Combinations

If \(\overline{a}=\alpha\overline{m}+\beta\overline{n}\) and \(\overline{b}=l\overline{m}+d\overline{n}\), where \(|\overline{m}|,\,|\overline{n}|\) and the angle \(\widehat{\overline{m},\overline{n}}\) are known and \(\alpha,\beta,l,d=\text{const}\), then

I.4 (c).Angle Between Two Vectors

The angle \(\varphi\) between vectors \(\overline{a}\) and \(\overline{b}\) is computed by the formula

I.4 (d).Angle Between Two Vectors in Coordinates

If \(\overline{a}=\{x_1,y_1,z_1\}\) and \(\overline{b}=\{x_2,y_2,z_2\}\), then

I.5.Vector Product of Vectors

Definition. The vector product of a vector \(\overline{a}\) by a vector \(\overline{b}\) is the vector \(\overline{c}\) satisfying the following conditions:

1. \(|\overline{c}|=|\overline{a}|\cdot|\overline{b}|\sin\bigl(\widehat{\overline{a},\overline{b}}\bigr)\);
2. \(\overline{c}\perp\overline{a},\ \overline{c}\perp\overline{b}\);
3. the vector \(\overline{c}\) is directed so that, seen from its end, the shortest rotation from \(\overline{a}\) to \(\overline{b}\) is counterclockwise.

The vector product is denoted \(\overline{a}\times\overline{b}\) or \([\overline{a}\cdot\overline{b}]\).

Properties of the vector product:

1. \(\overline{a}\times\overline{b}=-\,\overline{b}\times\overline{a}\) — anticommutative;
2. \(\overline{a}\times\overline{b}=0\) if \(\overline{a}=0\) or \(\overline{b}=0\), or \(\overline{a}\parallel\overline{b}\);
3. \((\lambda\overline{a})\times\overline{b}=\overline{a}\times(\lambda\overline{b})=\lambda(\overline{a}\times\overline{b})\) — the scalar factor may be taken outside;
4. \(\overline{a}\times(\overline{b}+\overline{c})=\overline{a}\times\overline{b}+\overline{a}\times\overline{c}\) — distributive over addition.

If \(\overline{a}=\{x_1,y_1,z_1\}\) and \(\overline{b}=\{x_2,y_2,z_2\}\), then

I.5 (a).Area of a Triangle

The area of the triangle built on the vectors \(\overline{a}=\{x_1,y_1,z_1\}\) and \(\overline{b}=\{x_2,y_2,z_2\}\) is computed by the formula

I.5 (b).Vector Product of Two Combinations · Area of a Parallelogram

If \(\overline{a}=\alpha\overline{m}+\beta\overline{n}\) and \(\overline{b}=l\overline{m}+d\overline{n}\), where \(\alpha,\beta,l,d=\text{const}\), then

since \([\overline{m},\overline{n}]=-[\overline{n},\overline{m}]\); \([\overline{m},\overline{m}]=0\); \([\overline{n},\overline{n}]=0\). The area of the parallelogram is

I.5 (c).A Vector Perpendicular to Two Vectors

If \(\overline{a}=\{x_1,y_1,z_1\}\), \(\overline{b}=\{x_2,y_2,z_2\}\) and \(\overline{c}\perp\overline{a},\ \overline{c}\perp\overline{b}\), then

or

where

I.6.Mixed Product of Three Vectors

Definition. The mixed product of vectors \(\overline{a},\overline{b}\) and \(\overline{c}\) is the scalar product of the vector \(\overline{a}\times\overline{b}\) with the vector \(\overline{c}\), i.e. \((\overline{a}\times\overline{b})\cdot\overline{c}\).

Properties of the mixed product:

1. The mixed product equals zero if: (a) at least one of the factors is zero; (b) two of the factors are collinear (lie on parallel lines); (c) the three nonzero vectors are parallel to one and the same plane (coplanarity).
2. The mixed product does not change if the signs of the vector \((\times)\) and scalar \((\cdot)\) products are interchanged, i.e. \((\overline{a}\times\overline{b})\cdot\overline{c}=\overline{a}\cdot(\overline{b}\times\overline{c})\); it is therefore written simply \(\overline{a}\,\overline{b}\,\overline{c}\).
3. It does not change under a cyclic permutation of the vectors: \(\overline{a}\,\overline{b}\,\overline{c}=\overline{b}\,\overline{c}\,\overline{a}=\overline{c}\,\overline{a}\,\overline{b}\).
4. Swapping any two vectors changes only its sign: \(\overline{b}\,\overline{a}\,\overline{c}=-\overline{a}\,\overline{b}\,\overline{c};\ \ \overline{c}\,\overline{b}\,\overline{a}=-\overline{a}\,\overline{b}\,\overline{c};\ \ \overline{a}\,\overline{c}\,\overline{b}=-\overline{a}\,\overline{b}\,\overline{c}.\)

If \(\overline{a}=\{x_1,y_1,z_1\}\); \(\overline{b}=\{x_2,y_2,z_2\}\); \(\overline{c}=\{x_3,y_3,z_3\}\), then

From these properties it follows that:

(a) the necessary and sufficient condition for the coplanarity of three vectors is \(\overline{a}\,\overline{b}\,\overline{c}=0\), i.e.

(b) the volume \(V_1\) of the parallelepiped built on \(\overline{a},\overline{b},\overline{c}\) and the volume \(V_2\) of the triangular pyramid (tetrahedron) they form are found by the formulas \(V_1=\bigl|\overline{a}\,\overline{b}\,\overline{c}\bigr|\), or

or

I.7.Projection of a Vector \(\overline{AB}\) onto an Axis \(l\)

The projection of a vector \(\overline{AB}\) onto an axis \(l\) is the value of the directed segment \(A'B'\) enclosed between the projections of the start and the end of \(\overline{AB}\), taken with a positive sign when \(\overline{A'B'}\) has the direction of the ort of the axis \(l\), and with a negative sign when \(\overline{A'B'}\) and the ort have opposite directions:

Necessary Information

On the topic “Curves of the Second Order” you should know:

• the definitions of the ellipse, the hyperbola and the parabola;
• the canonical equations of these curves;
• the geometric meaning of the parameters entering the equations, and the relations between the parameters,

and be able to:

• determine the type of a curve from its equation;
• derive the canonical equations of second-order curves;
• find the coordinates of the foci and the other parameters of the curves, and construct the curve.

I.Definition and Canonical Equation

Definition. An ellipse is the set of points of the plane, the sum of whose distances to two given points — called the foci \(F_1\) and \(F_2\) — is a constant (denoted by \(2a\)). Moreover this constant is greater than the distance between the foci.

If the coordinate axes are placed relative to the ellipse as shown in Fig. 1, and the foci lie on the \(Ox\) axis at equal distances from the origin at the points \(F_1(c;0)\), \(F_2(-c;0)\), one obtains the simplest (canonical) equation of the ellipse:

where \(a\) is the semi-major and \(b\) the semi-minor axis of the ellipse, and \(a,\,b\) and \(c\) (\(c\) is half the distance between the foci) are related by \(a^2=b^2+c^2\).

The shape of the ellipse (its measure of “flattening”) is characterised by its eccentricity: \(\varepsilon=\dfrac{c}{a}\) (since \(c<a\), we have \(\varepsilon<1\)).

The lines \(D_1:\ x=-\dfrac{a}{\varepsilon}\) and \(D_2:\ x=+\dfrac{a}{\varepsilon}\), perpendicular to the major axis and passing at distance \(\dfrac{a}{\varepsilon}\) from the centre, are called the directrices of the ellipse.

I.1.Location of the Ellipse and Its Parameters

1) \(a>b,\ c=\sqrt{a^2-b^2},\ F_1(x_0+c;\,y_0),\ F_2(x_0-c;\,y_0)\);

2) \(a<b,\ c=\sqrt{b^2-a^2},\ F_1(x_0;\,y_0+c),\ F_2(x_0;\,y_0-c)\).

I.2.Eccentricity

1) \(a>b;\ \varepsilon=\dfrac{c}{a}=\dfrac{\sqrt{a^2-b^2}}{a}\) — eccentricity;

2) \(a<b;\ \varepsilon=\dfrac{c}{b}=\dfrac{\sqrt{b^2-a^2}}{b}\) — eccentricity.

I.3.Equations of the Directrices

1) \(a>b\): \(\quad D_2:\ x=x_0+\dfrac{a}{\varepsilon};\qquad D_1:\ x=x_0-\dfrac{a}{\varepsilon}\);

2) \(b>a\): \(\quad D_2:\ y=y_0+\dfrac{b}{\varepsilon};\qquad D_1:\ y=y_0-\dfrac{b}{\varepsilon}\).

II.Definition and Canonical Equation

Definition. A hyperbola is the set of points of the plane, the absolute value of the difference of whose distances to two given points — called the foci — is a constant (denoted by \(2a\)); moreover this constant is less than the distance between the foci.

If the foci of the hyperbola are placed at the points \(F_1(c;0)\) and \(F_2(-c;0)\), one obtains the canonical equation of the hyperbola

The points \(A_1(a;0)\) and \(A_2(-a;0)\) are called the vertices of the hyperbola. The segment \(A_1A_2\) with \(|A_1A_2|=2a\) is called the real (transverse) axis of the hyperbola, and the segment \(B_1B_2\) with \(|B_1B_2|=2b\) the imaginary axis.

The hyperbola has two asymptotes, whose equations are

The ratio \(\varepsilon=\dfrac{c}{a}>1\) is called the eccentricity of the hyperbola.

The equation \(\dfrac{y^2}{b^2}-\dfrac{x^2}{a^2}=1\) is also an equation of a hyperbola, but the real axis of this hyperbola is the segment of the \(Oy\) axis of length \(2b\).

The two hyperbolas \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\) and \(\dfrac{y^2}{b^2}-\dfrac{x^2}{a^2}=1\) have the same semi-axes and the same asymptotes, but the real axis of one serves as the imaginary axis of the other and vice versa. Such two hyperbolas are called conjugate.

The lines \(D_1:\ x=-\dfrac{a}{\varepsilon}\) and \(D_2:\ x=+\dfrac{a}{\varepsilon}\), perpendicular to the real axis and passing at distance \(\dfrac{a}{\varepsilon}\) from the centre, are called the directrices of the hyperbola.

II.1.Location of the Hyperbola

1) \(\dfrac{(x-x_0)^2}{a^2}-\dfrac{(y-y_0)^2}{b^2}=1;\ O'(x_0;y_0)\) — centre, \(c=\sqrt{a^2+b^2}\), \(F_1(x_0+c;\,y_0)\), \(F_2(x_0-c;\,y_0)\) (Fig. 5);

2) \(\dfrac{(y-y_0)^2}{b^2}-\dfrac{(x-x_0)^2}{a^2}=1;\ O'(x_0;\,y_0)\) — centre, \(c=\sqrt{a^2+b^2}\), \(F_1(x_0;\,y_0-c)\), \(F_2(x_0;\,y_0+c)\) (Fig. 6).

II.2.Eccentricity

1) \(\dfrac{(x-x_0)^2}{a^2}-\dfrac{(y-y_0)^2}{b^2}=1;\ \varepsilon=\dfrac{c}{a}=\dfrac{\sqrt{a^2+b^2}}{a}\) — eccentricity.

2) \(\dfrac{(y-y_0)^2}{b^2}-\dfrac{(x-x_0)^2}{a^2}=1;\ \varepsilon=\dfrac{c}{b}=\dfrac{\sqrt{a^2+b^2}}{b}\) — eccentricity.

II.3.Equations of the Directrices of the Hyperbola

1) \(\dfrac{(x-x_0)^2}{a^2}-\dfrac{(y-y_0)^2}{b^2}=1,\)

2) \(\dfrac{(y-y_0)^2}{b^2}-\dfrac{(x-x_0)^2}{a^2}=1,\)

II.4.Equation of the Asymptotes

III.Definition and Canonical Equation

Definition. A parabola is the set of points of the plane equidistant from a given point — called the focus — and a given line — called the directrix.

If the directrix of the parabola is the line \(D:\ x=-\dfrac{p}{2}\), and the focus is the point \(F\left(\dfrac{p}{2};\,0\right)\), then the equation of the parabola has the form

III.1.Location of the Parabola and Its Parameters

1) \((x-x_0)^2=2p(y-y_0);\ O'(x_0,y_0)\) — vertex, \(p>0,\ F(x_0,\,y_0+\tfrac{p}{2})\) (Fig. 8) or \(p<0,\ F(x_0,\,y_0+\tfrac{p}{2})\) (Fig. 9).

2) \((y-y_0)^2=2p(x-x_0);\ O'(x_0,y_0)\) — vertex, \(p>0,\ F(x_0+\tfrac{p}{2},\,y_0)\) (Fig. 10) or \(p<0,\ F(x_0+\tfrac{p}{2},\,y_0)\) (Fig. 11).

III.2.Equation of the Directrix of the Parabola

1) \((x-x_0)^2=2p(y-y_0),\)

2) \((y-y_0)^2=2p(x-x_0),\)

IV.1.Type from the Product \(A\cdot C\)

For the investigation of second-order curves, whose general equation has the form \(Ax^2+Cy^2+2Dx+2Ey+F=0\), one considers the product \(A\cdot C\):

• if \(A\cdot C>0\), we have an ellipse;
• if \(A\cdot C<0\), we have a hyperbola;
• if \(A\cdot C=0\), we have a parabola.

IV.2.Completing the Square

In the investigation of second-order curves whose equation is written in general form, the “procedure of completing the square” is useful. Completing the square of the equation \(Ax^2+Cy^2+2Dx+2Ey+F=0\), we obtain

or

Denote: \(x_0=-\dfrac{D}{A},\ y_0=-\dfrac{E}{C}\),

IV.2 (a).Elliptic Type \((A\cdot C>0)\)

If \(A\cdot C>0\), the equation defines a curve of elliptic type. Moreover:

a) if \(\dfrac{D^2}{A}+\dfrac{E^2}{C}-F<0\), we have an imaginary ellipse;

b) if \(\dfrac{D^2}{A}+\dfrac{E^2}{C}-F=0\), we have the point \(\left(-\dfrac{D}{A};\,-\dfrac{E}{C}\right)\);

c) if \(\dfrac{D^2}{A}+\dfrac{E^2}{C}-F>0\), then

and we have the canonical form of an ellipse.

IV.2 (b).Hyperbolic Type \((A\cdot C<0)\)

If \(A\cdot C<0\), the equation defines a curve of hyperbolic type. Moreover:

a) if \(\dfrac{D^2}{A}+\dfrac{E^2}{C}-F>0\) or \(\dfrac{D^2}{A}+\dfrac{E^2}{C}-F<0\), then

i.e. the canonical form of a hyperbola;

b) if \(\dfrac{D^2}{A}+\dfrac{E^2}{C}-F=0\), then, taking the signs of \(A\) and \(C\) into account, we have \(y-y_0=\pm\dfrac{b}{a}(x-x_0)\), i.e. a pair of intersecting lines.

IV.2 (c).Parabolic Type

a) If \(C=0\), the general equation \(Ax^2+Cy^2+2Dx+2Ey+F=0\) defines a curve of parabolic type. Completing the square, we have:

Denote \(x_0=-\dfrac{D}{A},\ y_0=-\dfrac{F-\dfrac{D^2}{A}}{2E},\ p=-\dfrac{E}{A}\). Then \((x-x_0)^2=2p(y-y_0)\), i.e. the canonical form of a parabola;

b) if \(A=0\), then \(Ax^2+Cy^2+2Dx+2Ey+F=0\), i.e. a curve of parabolic type. Completing the square, we have:

Denote \(x_0=-\dfrac{F-\dfrac{E^2}{C}}{2D},\ y_0=-\dfrac{E}{C},\ p=-\dfrac{D}{C}\). Then \((y-y_0)^2=2p(x-x_0)\), i.e. the canonical form of a parabola.

V.1.Equations of the Form \(y=b+\sqrt{ax^2+cx+d}\)

Consider an equation of the form \(y=b+\sqrt{ax^2+cx+d}\). This equation is equivalent to the system

V.1 (a). If \(d-\dfrac{c^2}{4a}>0\) and \(a>0\), the equation \((y-b)^2-a\left(x+\tfrac{c}{2a}\right)^2=d-\tfrac{c^2}{4a}\) defines the part of a hyperbola lying in the half-plane \(y\ge b\) (Fig. 12). We construct the part of the hyperbola above the line \(y=b\).

V.1 (b). If \(d-\dfrac{c^2}{4a}>0\) and \(a<0\), the equation defines the part of an ellipse lying in the half-plane \(y\ge b\) (Fig. 13). We construct the part of the ellipse above the line \(y=b\).

V.1 (c). If \(d-\dfrac{c^2}{4a}<0\) and \(a>0\), the equation defines the part of a hyperbola lying in the half-plane \(y\ge b\) (Fig. 14).

V.1 (d). If \(d-\dfrac{c^2}{4a}<0\) and \(a<0\), the equation defines an imaginary ellipse.

V.1 (e). If \(a=0\), the equation \(y=b+\sqrt{cx+d}\) is equivalent to the system \(\begin{cases}y-b\ge 0\\ (y-b)^2=c\left(x+\tfrac{d}{c}\right)\end{cases}\), which defines the part of a parabola lying in the half-plane \(y\ge b\) (Figs. 15, 16).

V.1 (f). If \(d-\dfrac{c^2}{4a}=0\) and \(a>0\), the equation defines a pair of intersecting lines \(y-b=\pm\sqrt{a}\left(x+\tfrac{c}{2a}\right)\), which lie in the half-plane \(y\ge b\).

V.1 (g). If \(d-\dfrac{c^2}{4a}=0\) and \(a<0\), the equation defines the point \(\left(-\tfrac{c}{2a};\,b\right)\), provided it lies in the half-plane \(y\ge b\).

V.2.Equations of the Form \(y=b-\sqrt{ax^2+cx+d}\)

Consider an equation of the form \(y=b-\sqrt{ax^2+cx+d}\). This equation is equivalent to the system

since only positive values of the root are considered.

V.2 (a). If \(d-\dfrac{c^2}{4a}>0\) and \(a>0\), the equation defines the part of a hyperbola lying in the half-plane \(y\le b\) (Fig. 17).

V.2 (b). If \(d-\dfrac{c^2}{4a}>0\) and \(a<0\), the equation defines the part of an ellipse lying in the half-plane \(y\le b\) (Fig. 18).

V.2 (c). If \(d-\dfrac{c^2}{4a}<0\) and \(a>0\), the equation defines the part of a hyperbola lying in the half-plane \(y\le b\) (Fig. 19).

V.2 (d). If \(d-\dfrac{c^2}{4a}<0\) and \(a<0\), the equation defines an imaginary ellipse.

V.2 (e). If \(a=0\), the equation \(y=b-\sqrt{cx+d}\) is equivalent to the system \(\begin{cases}y\le b\\ (y-b)^2=c\left(x+\tfrac{d}{c}\right)\end{cases}\), which defines the part of a parabola lying in the half-plane \(y\le b\) (Figs. 20, 21).

V.2 (f). If \(d-\dfrac{c^2}{4a}=0\) and \(a>0\), the equation defines a pair of intersecting lines \(y-b=\pm\sqrt{a}\left(x+\tfrac{c}{2a}\right)\), which lie in the half-plane \(y\le b\).

V.2 (g). If \(d-\dfrac{c^2}{4a}=0\) and \(a<0\), the equation defines the point \(\left(-\tfrac{c}{2a};\,b\right)\), provided it lies in the half-plane \(y\le b\).