Question

Match options in List 1 with options in List 2

Answer 1

1) $hx+ky-1=0$ touches the circle $x^2+y^2=4$
$\Rightarrow$ $\frac{|h\left(0\right)+k\left(0\right)-1|}{\sqrt{h^2+k^2}}=4\Rightarrow h^2+k^2=1/4$ and the locus of $(h,k)$ is $x^2+y^2=1/4$ which is a circle.
2) Since the difference of the distance of the point $z$ from two fixed points $(2,0)$ and $(-2,0)$ is constant. Its locus is a Hyperbola.
3) ${\left(\frac{x}{\sqrt{3}}\right)}^2+y^2=\frac{{(1-t^2)}^2}{{(1+t^2)}^2}+\frac{{\left(2t\right)}^2}{{(1+t^2)}^2}=1$
$\Rightarrow$ $\frac{x^2}{3}+y^2=1$ which represents an ellipse.
4)For eccentricity $e=1$, conic is a parabola and for $e>1$ conic is a 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 a parabola.