Question

Statement-1: Let the mapping $w = z +\frac{ c }{ z }$ where $z = x + iy , w = u + iv$ and $c$ is a real number $(\neq 0, \pm 1)$ maps the circle $|z|=1$ in the $z$ plane into a conic in the $w$ plane. The conic is a hyperbola.
Statement-2: The equation $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ represents a hyperbola provided that $a b c+2 f g h-a f^2-b g^2-c h^2 \neq 0$ and $h^2-a b>0$.

• A. Statement- 1 is true, statement -2 is true and statement- 2 is correct explanation for statement- 1 .
• B. Statement- 1 is true, statement- 2 is true and statement- 2 is NOT the correct explanation for statement- 1 .
• C. Statement- 1 is true, statement- 2 is false.
• D. Statement- 1 is false, statement- 2 is true.

Answer 1

Answer: (D)

We have $u+i v=z+\frac{c}{z} \Rightarrow u+i v=(x+i y)+\frac{c}{x+i y} \Rightarrow u+i v=(x+i y)+\frac{c(x-i y)}{x^2+y^2}$ $\Rightarrow u + iv = x (1+ c )+ i (1- c ) y \left(\right.$ using $\left.x ^2+ y ^2=1\right)$
Hence $\frac{ u ^2}{(1+ c )^2}+\frac{ v ^2}{(1- c )^2}= x ^2+ y ^2=1$, which is an ellipse.