IIT-JEE 2009 · Paper 1 · Mathematics Q40

MathematicsMatrix match+8Question 40 of 60

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

Column I

Column II

P
The locus of the point $(h,k)$ for which the line $hx+ky=1$ touches the circle $x^2+y^2=4$
Q
Points $z$ in the complex plane satisfying $|z+2|-|z-2|=\pm3$
R
Points of the conic have parametric representation $x=\sqrt3\left(\dfrac{1-t^2}{1+t^2}\right)$ , $y=\dfrac{2t}{1+t^2}$
S
The eccentricity of the conic lies in the interval $1\le x<\infty$
T
Points $z$ in the complex plane satisfying $\operatorname{Re}(z+1)^2=|z|^2+1$

Answer and worked solution

Official answer: A → P; B → S, T; C → R; D → Q, S

P: $hx+ky=1$ touches $x^2+y^2=4$ when $\dfrac1{\sqrt{h^2+k^2}}=2$ : $h^2+k^2=\frac14$ , a circle.
Q: $\big||z+2|-|z-2|\big|=3<4$ : a hyperbola with foci $\pm2$ .
R: $x=\sqrt3\cos\phi$ , $y=\sin\phi$ with $t=\tan\frac\phi2$ : the ellipse $\frac{x^2}3+y^2=1$ .
S: eccentricity at least 1: parabola ( $e=1$ ) or hyperbola ( $e>1$ ).
T: $(x+1)^2-y^2=x^2+y^2+1$ gives $y^2=x$ , a parabola.

Answer: A → P; B → S, T; C → R; D → Q, S

IIT-JEE 2009 · Paper 1, Mathematics question 40