Question

Statement-1: Let the mapping $w=z+\frac{c}{z}$ where $z=x+iy,w=u+iv$ and $c$ is a real number $(\ne 0,\pm 1)$ 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 $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a hyperbola provided that $abc+2fgh-a{f}^2-b{g}^2-c{h}^2\ne 0$ and $h^2-ab>0$.

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