IIT JEE Advanced 2009 Paper 1

(P) We have $\frac{1}{k^2}$ = $4\left(1+\frac{h^2}{k^2}\right)$ ⇒ 1 = $4(k^2+h^2)$ Hence, $h^2+k^2$ = $\left(\frac{1}{2}\right)^2$, which is a circle. (Q) If $\left|z-z_1\right|-\left|z-z_2\right|$ = k, where < $\left|z_1-z_2\right|$, the locus is a hyperbola. (R) Let t = tan α. Hence, x = $\sqrt{3}\cos 2\alpha$ and y = sin 2α or cos 2α = $\frac{x}{\sqrt{3}}$ and sin 2α = y $\frac{x^3}{3}+y^2$ = $\sin^2 2\alpha + \cos^2 2\alpha$ = 1, which is an ellipse (S) If eccentricity is [1, ∞), then the conic can be a parabola (if e = 1) and a hyperbola if e ∈ (1,∞). (T) Let z = x + iy; x, y ∈ R. Hence, $(x+1)^2-y^2$ = $x^2+y^2+1$ ⇒ $y^2$ = x; which is a parabola.