1
IIT-JEE 2009 Paper 2 Offline
+3
-1
The normal at a point $$P$$ on the ellipse $${x^2} + 4{y^2} = 16$$ meets the $$x$$- axis $$Q$$. If $$M$$ is the mid point of the line segment $$PQ$$, then the locus of $$M$$ intersects the latus rectums of the given ellipse at the points
A
$$\left( { \pm {{3\sqrt 5 } \over 2},\, \pm {2 \over 7}} \right)$$
B
$$\left( { \pm {{3\sqrt 5 } \over 2},\, \pm \sqrt {{{19} \over 4}} } \right)$$
C
$$\left( { \pm 2\sqrt 3 , \pm {1 \over 7}} \right)$$
D
$$\left( { \pm 2\sqrt 3 , \pm {{4\sqrt 3 } \over 7}} \right)$$
2
IIT-JEE 2009 Paper 2 Offline
+3
-1

The locus of the orthocentre of the triangle formed by the lines

$$(1 + p)x - py + p(1 + p) = 0,$$

$$(1 + q)x - qy + q(1 + q) = 0$$

and $$y = 0$$, where $$p \ne q$$, is :

A
a hyperbola.
B
a parabola.
C
an ellipse.
D
a straight line.
3
IIT-JEE 2009 Paper 1 Offline
+3
-1
The line passing through the extremity $$A$$ of the major axis and extremity $$B$$ of the minor axis of the ellipse $${x^2} + 9{y^2} = 9$$ meets its auxiliary circle at the point $$M$$. Then the area of the triangle with vertices at $$A$$, $$M$$ and the origin $$O$$ is
A
$${{31} \over {10}}$$
B
$${{29} \over {10}}$$
C
$${{21} \over {10}}$$
D
$${{27} \over {10}}$$
4
IIT-JEE 2009 Paper 1 Offline
+3
-0

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