1
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $${E_1}$$ and $${E_2}$$ be two ellipses whose centres are at the origin. The major axes of $${E_1}$$ and $${E_2}$$ lie along the $$x$$-axis and the $$y$$-axis, respectively. Let $$S$$ be the circle $${x^2} + {\left( {y - 1} \right)^2} = 2$$. The straight line $$x+y=3$$ touches the curves $$S$$, $${E_1}$$ and $${E_2}$$ at $$P, Q$$ and $$R$$ respectively. Suppose that $$PQ = PR = {{2\sqrt 2 } \over 3}$$. If $${e_1}$$ and $${e_2}$$ are the eccentricities of $${E_1}$$ and $${E_2}$$, respectively, then the correct expression(s) is (are)
A
$$\mathop e\nolimits_1^2 + \mathop e\nolimits_2^2 = {{43} \over {40}}$$
B
$${e_1}{e_2} = {{\sqrt 7 } \over {2\sqrt {10} }}$$
C
$$\left| {\mathop e\nolimits_1^2 + \mathop e\nolimits_2^2 } \right| = {5 \over 8}$$
D
$${e_1}{e_2} = {{\sqrt 3 } \over 4}$$
2
IIT-JEE 2009 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
An ellipse intersects the hyperbola $$2{x^2} - 2{y^2} = 1$$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes then
A
equation of ellipse is $${x^2} + 2{y^2} = 2$$
B
the foci of ellipse are $$\left( { \pm 1,0} \right)$$
C
equation of ellipse is $${x^2} + 2{y^2} = 4$$
D
the foci of ellipse are $$\left( { \pm \sqrt 2 ,0} \right)$$
3
IIT-JEE 2009 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
In a triangle $$ABC$$ with fixed base $$BC$$, the vertex $$A$$ moves such that $$\cos \,B + \cos \,C = 4{\sin ^2}{A \over 2}.$$\$

If $$a, b$$ and $$c$$ denote the lengths of the sides of the triangle opposite to the angles $$A, B$$ and $$C$$, respectively, then

A
$$b+c=4a$$
B
$$b+c=2a$$
C
locus of point $$A$$ is an ellipse
D
locus of point $$A$$ is a pair of straight lines
4
IIT-JEE 2008 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
Let $$P\left( {{x_1},{y_1}} \right)$$ and $$Q\left( {{x_2},{y_2}} \right),{y_1} < 0,{y_2} < 0,$$ be the end points of the latus rectum of the ellipse $${x^2} + 4{y^2} = 4.$$ The equations of parabolas with latus rectum $$PQ$$ are :
A
$${x^2} + 2\sqrt 3y = 3 + \sqrt 3$$
B
$${x^2} - 2\sqrt 3y = 3 + \sqrt 3$$
C
$${x^2} + 2\sqrt 3y = 3 - \sqrt 3$$
D
$${x^2} - 2\sqrt 3 y = 3 - \sqrt 3$$
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