Question

Match the following equations with corresponding conic:

• A. Circle
• B. Straight Line
• C. Parabola

Answer 1

We have to decide the curve represented by the equations given. The systematic way of finding the conic is as follows.

Let the equation of the curve be

$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ and

$△=abc+2fgh-a{f}^2-b{g}^2-c{h}^2$

1) Find $△$

2) If $△=0$, it represents a pair of straight lines.

3) If $△\ne 0$, it is a conic.

To decide the type of conic, we will compare $h^2$ and $ab.$

4) If $h^2=ab\Rightarrow$ parabola.

5) If $h^2<ab\Rightarrow$ ellipse or circle.

For circle, $a=b$ and $h=0$

6) If $h^2>ab\Rightarrow$ hyperbola.

We will go through each equation and decide the conic it represents.

(I) $x^2-2x-y=0$

$a=1,b=0,h=0,g=-1,f=-\frac{1}{2}$

$△=0+0-1\times {\left(\frac{-1}{2}\right)}^2-0-0=\frac{-1}{4}\ne 0$
$\Rightarrow$ It is the equation of a conic.

Now, $h^2=0$ and $ab=0$

$\because h^2=ab\Rightarrow$ It is a parabola.

(II) $x^2+y^2-20=0$

If $a=b$ and $h=0,$ it represents a circle.

Here, $a=b=1$ and $h=0\Rightarrow$ It is a circle.

(III) Clearly, $x+2y+3=0$ is a linear equation which is the equation of a straight line.