Question

The line $$y=x$$ intersects the hyperbola $$\dfrac{x^2}{9}-\dfrac{y^2}{25}=1$$ at the points P and Q. The eccentricity of ellipse with PQ as major axis and minor axis of length $$\dfrac{5}{\sqrt{2}}$$ is

A
53
B
53
C
59
D
259

Solution

The correct option is A $$\dfrac{\sqrt{5}}{3}$$Line $$y=x$$ intersects hyperbola $$\dfrac { { x }^{ 2 } }{ { 9 } } -\dfrac { { y }^{ 2 } }{ 25 } =1$$ at P and Q putting $$y=x$$ in eq. of parabola     $$\dfrac { { x }^{ 2 } }{ { 9 } } -\dfrac { { y }^{ 2 } }{ 25 } =1$$     $$16{ y }^{ 2 }=9\times 25$$     $$y=\pm \dfrac { 3\times 5 }{ 4 }$$     $$x=\pm \dfrac { 3\times 5 }{ 4 }$$PQ = Distance b/w points $$\left( \dfrac { +15 }{ 4 } ,\dfrac { +15 }{ 4 } \right)$$ and $$\left( \dfrac { -15 }{ 4 } ,\dfrac { -15 }{ 4 } \right)$$      $$=\sqrt { { \left( \dfrac { 15 }{ 2 } \right) }^{ 2 }+{ \left( \dfrac { 15 }{ 2 } \right) }^{ 2 } }$$      $$=\dfrac { 15 }{ \sqrt { 2 } }$$Length of major axis $$\Rightarrow \dfrac { 15 }{ \sqrt { 2 } }$$, length of minor axis $$=\dfrac { 5 }{ \sqrt { 2 } }$$      $$e=\sqrt { 1-\dfrac { { \left( \dfrac { 5 }{ \sqrt { 2 } } \right) }^{ 2 } }{ { \left( \dfrac { 15 }{ \sqrt { 2 } } \right) }^{ 2 } } } \Rightarrow \quad e=\sqrt { \dfrac { 200 }{ 225 } } \Rightarrow \dfrac {{2} \sqrt { 2 } }{ 3 }$$Maths

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