Conic Sections - Practice Questions & MCQ

Introduction to Conic Section

The two-dimensional curves that can be generated as cross-sections when a double cone is cut by a plane at different angles are called conic sections. 

We obtain different kinds of conic sections depending on the position of the intersecting plane concerning the cone and by the angle made by it with the vertical axis of the cone.

Let β be the angle made by the intersecting plane with the vertical axis of the cone.

For example, parabola, ellipse and hyperbola are conic sections as they are formed when a plane cuts a double cone as shown in the figure.

Note: a circle is a special case of ellipse when the plane is perpendicular to the axis of the double cone, and hence a circle is also a conic section

Definition of Conic Section in 2-dimensions

A conic section is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed line.

The fixed point is called the Focus of the conic and this fixed line is called the Directrix of the conic.

Also, the constant ratio is called the Eccentricity and is denoted by e.

From the figure

Some Important Definitions to Remember

Axis: The straight line passing through the focus and perpendicular to the directrix is called the axis of the conic section.

Vertex: The point of intersection of the conic section and the axis is called the vertex of the conic section. 

Double ordinate: Any chord, which is perpendicular to the axis of the conic section, is called a double ordinate of the conic section. 

Focal chord: Any chord passing through the focus is called the focal chord of the conic section. 

Focal distance: The distance between the focus and any point on the conic is known as the focal distance of that point.

Latus rectum: Any chord passing through the focus and perpendicular to the axis is known as the latus rectum of the conic section. 

Centre: The point which bisects every chord of the conic passing through it, is called the centre of the conic section.

Equation of Conic Section:

The focus at $\mathrm{S}(\mathrm{h},\mathrm{k})$ and the Directrix : $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$, then equation of conix

$\begin{array}{rl}& \Rightarrow PS=e\cdot PM\\ & \Rightarrow \sqrt{(x-h{)}^2+(y-k{)}^2}=e\cdot \frac{|ax+by+c|}{\sqrt{a^2+b^2}}\\ & \Rightarrow (x-h{)}^2+(y-k{)}^2=e^2\cdot \frac{(ax+by+c{)}^2}{a^2+b^2}\end{array}$
 

Recognition of Conics

Equation of conic represented by the general equation of second-degree

$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$
The discriminant of the above equation is denoted by $\mathrm{\Delta }$, where

$\mathrm{\Delta }=abc+2fgh-a{f}^2-b{g}^2-c{h}^2$
It can also be written in determinant form as

$\left|\begin{array}{lll}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|$
Case 1: When the focus lies on the directrix
$\mathrm{\Delta }=0$ : In this case, the equation represents a pair of straight lines

Case 2: When the focus does not lie on the directrix