Conic Sections

Introduction

• When a solid surface like a cone is cut by a plane it forms different sections called conic sections.

• There are different conic sections like a circle, ellipse, parabola, and hyperbola based on the angle at which the plane intersects with the cone axis.

• Circle and ellipse are closed sections whereas parabola and hyperbola are not closed the curves of these figures extend till infinity.

Definition of Section of Cone

• When a two-dimensional plane intersects a three-dimensional right circular cone the crop section formed on the plane is called a conic section.

• The plane intersects the cone at different angles to form different conic sections like circle, parabola, ellipse, and hyperbola. There are different parameters and equations for different conic sections.

Parameters of a Conic Section

There are different parameters for a conic section other than focus, directrix, and eccentricity.

• Latus Rectum − It is a line parallel to the directrix, and also should pass through the focus of the conic section is called the latus rectum of a conic section.

• Focal Parameter − The distance from the focus of a conic section to its corresponding directrix is called the Focal Parameter.

• Principal Axis − The line which passes through focus and directrix for a parabola or the line joining the two foci for a hyperbola and ellipse is called a Principal Axis. Its midpoint is the centre of the conic section.

• Linear Eccentricity − The distance between the centre and focus of a conic section is called the Linear Eccentricity of the conic section.

• Major axis − The longest chord through the centre is called the Major axis.

• Minor axis − The smallest chord through the centre is called the Minor axis.

Focus Eccentricity and Directrix

• Directrix − The locus of a moving point in the plane of a fixed point and a fixed line is another way of defining a conic section. The fixed line is called the Directrix of a conic section.

• Focus − The fixed point is called the Focus of a conic section. The focus doesn't lie on the directrix of a conic section.

• Eccentricity − Consider a point P on the conic section. The ratio of the distance between point P to the focus and the perpendicular distance between point P to the directrix is called Eccentricity and it is constant for a given conic section. It is denoted by ‘e’.

• If the eccentricity of a conic section is zero, then it is a circle.

• If the eccentricity of a conic section strictly lies between zero and one, then it is an ellipse.

• If the eccentricity of a conic section is one, then it is a parabola.

• If the eccentricity of a conic section is greater than one, then it is a hyperbola.

Sections of Cone

The important conic sections are circle, parabola, ellipse, and hyperbola.

• Circle − When the plane cuts the cone at an angle of 90 degrees from the axis of the cone a conic section named a circle is formed.

• Ellipse − When the plane cuts the cone at an angle less than 90 degrees but more than the semi-vertical angle of the cone a conic section named an ellipse is formed.

• Parabola − When the plane cuts the cone at an angle equal to the semi- vertical angle of the cone a conic section named a parabola is formed.

• Hyperbola − When the plane cuts the cone at an angle less than the semi- vertical angle of the cone a conic section named a hyperbola is formed.

Equations of Sections of Cone

The four conic sections circle, ellipse, parabola, and hyperbola have their equations.

Circle

A circle is a locus of a set of points equally distanced from a point called the centre of the circle. The distance is called the radius of the circle. From the distance, the equation of the circle is derived.

Consider a circle with a centre (p, q) and a point P (x, y) lying on the circle. The radius of the circle is r.

The distance between point P and the centre is equal to radius r.

$\mathrm{(x\:-\:p)^{2}\:+\:(y\:-\:q)^{2}\:=\:r^{2}}$

Ellipse

An ellipse is a locus of a set of points whose sum of the distance from two fixed points called foci is the same.

Consider an ellipse with a centre (p, q), an arbitrary point (x, y) on the ellipse, the major axis as the x-axis, a is the semi-major axis, and b is the semi-minor axis.

$$\mathrm{\frac{(x\:-\:p)^{2}}{a^{2}}\:+\:\frac{(y\:-\:q)^{2}}{b^{2}}\:=\:1}$$

Parabola

A parabola is a locus of a set of points whose distance from the focus and the perpendicular distance from the directrix are equal.

Consider a parabola along the x-axis with centre (p, q), point P (x, y) on the parabola, and focus at (p + a, q).

$$\mathrm{(y\:-\:p)^{2}\:=4a\:(x\:-\:p)}$$

Hyperbola

A hyperbola is a locus of a set of points whose difference in the distance between two fixed points called foci is the same.

Consider a hyperbola along the x-axis with a centre (p, q), an arbitrary point (x, y) on the hyperbola, a is the semi-transverse axis, and b is the semi-conjugate axis.

$$\mathrm{\frac{(x\:-\:p)^{2}}{a^{2}}\:-\:\frac{(y\:-\:q)^{2}}{b^{2}}\:=\:1}$$

Examples

• What is the equation of the circle with a centre (2, 3) and a point P (x, y) lying on the circle given the radius of the circle is 4cm?

$\mathrm{(x\:-\:2)^{2}\:+\:(y\:-\:3)^{2}\:=\:4^{2}}$

• What is the centre, length of the semi-major axis, and length of the semi-minor axis of an ellipse whose equation is given as $\mathrm{\frac{(x\:-\:1)^{2}}{16}\:+\:\frac{(y\:-\:2)^{2}}{9}\:=\:1}$ ?

Comparing the given equation with $\mathrm{\frac{(x\:-\:p)^{2}}{a^{2}}\:+\:\frac{(y\:-\:q)^{2}}{b^{2}}\:=\:1}$ the centre of the ellipse is equal to (1, 2).

$\mathrm{a^{2}\:=\:16\Longrightarrow\:a\:=\:4\:,\:b^{2}\:=\:9\Longrightarrow\:b\:=\:3}$

Conclusion

In this tutorial, we learned about conic sections, their parameters like focus, eccentricity, directrix, different sections of a cone like circle, ellipse, parabola, hyperbola and their equations.

FAQs

1. What is the equation of a parabola along x-axis with origin as its centre?

$\mathrm{y^{2}\:=\:4ax}$ is the equation of the parabola with origin as its centre.

2. What is the equation of a circle with origin as its centre?

If origin is the centre of a circle is then the equation of the circle is $\mathrm{(x)^{2}\:+\:(y)^{2}\:=\:r^{2}}$

3. What is the condition of eccentricity for a conic section to be a hyperbola?

The eccentricity should be greater than one, for a conic section to be a hyperbola.

4. What is the equation of the hyperbola with a centre (-1, 4) and a point P (x, y) lying on the hyperbola given a = 5, and b = 3?

The equation of the hyperbola is $\mathrm{\frac{(x\:+\:1)^{2}}{25}\:-\:\frac{(y\:-\:4)^{2}}{9}\:=\:1}$

5. hat is the equation of a parabola along the y-axis with origin as its centre?

$\mathrm{x^{2}\:=\:4ay}$ is the equation of the parabola with origin as its centre.