JEE Advance - Mathematics (2009 - Paper 1 Offline - No. 10)

Match the conics in Column I with the statements/expressions in Column II :

	Column I		Column II
(A)	Circle	(P)	The locus of the point ($$h,k$$) for which the line $$hx+ky=1$$ touches the circle $$x^2+y^2=4$$.
(B)	Parabola	(Q)	Points z in the complex plane satisfying $$\|z+2\|-\|z-2\|=\pm3$$.
(C)	Ellipse	(R)	Points of the conic have parametric representation $$x = \sqrt 3 \left( {{{1 - {t^2}} \over {1 + {t^2}}}} \right),y = {{2t} \over {1 + {t^2}}}$$
(D)	Hyperbola	(S)	The eccentricity of the conic lies in the interval $$1 \le x \le \infty $$.
		(T)	Points z in the complex plane satisfying $${\mathop{\rm Re}\nolimits} {(z + 1)^2} = \|z{\|^2} + 1$$.

(A)$$\to$$(P); (B)$$\to$$(S), (T); (C)$$\to$$(R); (D)$$\to$$(R), (S)

(A)$$\to$$(P); (B)$$\to$$(S), (T); (C)$$\to$$(R); (D)$$\to$$(Q), (S)

(A)$$\to$$(P); (B)$$\to$$(S), (T); (C)$$\to$$(S); (D)$$\to$$(R), (S)

(A)$$\to$$(P); (B)$$\to$$(P), (T); (C)$$\to$$(R); (D)$$\to$$(Q), (S)

(P) We have $${1 \over {{k^2}}} = 4\left( {1 + {{{h^2}} \over {{k^2}}}} \right) \Rightarrow 1 = 4({k^2} + {h^2})$$

Hence, $${h^2} + {k^2} = {\left( {{1 \over 2}} \right)^2}$$ , which is a circle.

(Q) If $$|z - {z_1}| - |z - {z_2}| = k$$, where $$k < |{z_1} - {z_2}|$$, the locus is a hyperbola.

(R) Let $$t = \tan \alpha $$. Hence,

$$x = \sqrt 3 \cos 2\alpha $$ and $$y = \sin 2\alpha $$

or $$\cos 2\alpha = {x \over {\sqrt 3 }}$$ and $$\sin 2\alpha = y$$

$${{{x^2}} \over 3} + {y^2} = {\sin ^2}2\alpha + {\cos ^2}2\alpha = 1$$, which is an ellipse.

(S) If eccentricity is $$[1,\infty]$$, then the conic can be a parabola (if $$e=1$$) and a hyperbola if $$e\in(1,\infty)$$.

(T) Let $$z = x + iy;x,y \in R$$. Hence,

$${(x + 1)^2} - {y^2} = {x^2} + {y^2} + 1$$

$$ \Rightarrow {y^2} = x;$$ which is a parabola