1
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Consider the hyperbola $$H:{x^2} - {y^2} = 1$$ and a circle $$S$$ with center $$N\left( {{x_2},0} \right)$$. Suppose that $$H$$ and $$S$$ touch each other at a point $$P\left( {{x_1},{y_1}} \right)$$ with $${{x_1} > 1}$$ and $${{y_1} > 0}$$. The common tangent to $$H$$ and $$S$$ at $$P$$ intersects the $$x$$-axis at point $$M$$. If $$(l, m)$$ is the centroid of the triangle $$PMN$$, then the correct expressions(s) is(are)
A
$${{dl} \over {d{x_1}}} = 1 - {1 \over {3x_1^2}}$$ for $${x_1} > 1$$
B
$${{dm} \over {d{x_1}}} = {{{x_1}} \over {3\left( {\sqrt {x_1^2 - 1} } \right)}}$$ for $${x_1} > 1$$
C
$${{dl} \over {d{x_1}}} = 1 + {1 \over {3x_1^2}}$$ for $${x_1} > 1$$
D
$${{dm} \over {d{y_1}}} = {1 \over 3}$$ for $${y_1} > 0$$
2
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$P$$ and $$Q$$ be distinct points on the parabola $${y^2} = 2x$$ such that a circle with $$PQ$$ as diameter passes through the vertex $$O$$ of the parabola. If $$P$$ lies in the first quadrant and the area of the triangle $$\Delta OPQ$$ is $${3\sqrt 2 ,}$$ then which of the following is (are) the coordinates of $$P$$?
A
$$\left( {4,2\sqrt 2 } \right)$$
B
$$\left( {9,3\sqrt 2 } \right)$$
C
$$\left( {{1 \over 4},{1 \over {\sqrt 2 }}} \right)$$
D
$$\left( {1,\sqrt 2 } \right)$$
3
IIT-JEE 2012 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Tangents are drawn to the hyperbola $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1,$$ parallel to the straight line $$2x - y = 1,$$ The points of contact of the tangents on the hyperbola are
A
$$\left( {{9 \over {2\sqrt 2 }},{1 \over {\sqrt 2 }}} \right)$$
B
$$\left( -{{9 \over {2\sqrt 2 }},-{1 \over {\sqrt 2 }}} \right)$$
C
$$\left( {3\sqrt 3 , - 2\sqrt 2 } \right)$$
D
$$\left( -{3\sqrt 3 , 2\sqrt 2 } \right)$$
4
IIT-JEE 2011 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let the eccentricity of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be reciprocal to that of the ellipse $${x^2} + 4{y^2} = 4$$. If the hyperbola passes through a focus of the ellipse, then
A
the equation of the hyperbola is $${{{x^2}} \over 3} - {{{y^2}} \over 2} = 1$$
B
a focus of the hyperbola is $$(2, 0)$$
C
theeccentricity of the hyperbola is $$\sqrt {{5 \over 3}} $$
D
The equation of the hyperbola is $${x^2} - 3{y^2} = 3$$
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