Question

Match the conics in List 1 with the statements/expressions in List 2.

Answer 1

1) $hx+ky-1=0$ touches the circle $x^2+y^2=4$
$\Rightarrow \frac{|h\left(0\right)+k\left(0\right)-1|}{\sqrt{h^2+k^2}}=4\Rightarrow h^2+k^2=\frac{1}{4}$
and the locus of $(h,k)$ is $x^2+y^2=\frac{1}{4}$ i.e circle
2) Since the difference of the distance of the point $Z$ from two fixed point $(2,0)$ and $(-2,0)$ is constant.
$|z-z_1|-|z-z_2|=k$ where $k<|z_1-z_2|$
Then its locus is Hyperbola
3) Let $t=\tan \alpha \Rightarrow x=\sqrt{3}\cos 2\alpha ,y=\sin 2\alpha$
$\Rightarrow \cos 2\alpha =\frac{x}{\sqrt{3}},\sin 2\alpha =y\frac{x^2}{3}+y^2={\sin }^{2}2\alpha +{\cos }^{2}2\alpha =1$
It represent a ellipse
4) For eccentricity $e=1$, conic is a parabola and for $e>1$ conic is hyperbola
5) Let $z=x+iy$, then
$Re{(z+1)}^2={\left|z\right|}^2+1\Rightarrow (x+1{)}^2-y^2=x^2+y^2+1\Rightarrow y^2=x$
Which is parabola