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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
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B
53
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C
59
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D
259
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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 $$=\dfrac { 5 }{ \sqrt { 2 }  } $$
      $$e=\sqrt { 1-\dfrac { { \left( \dfrac { 5 }{ \sqrt { 2 }  }  \right)  }^{ 2 } }{ { \left( \dfrac { 15 }{ \sqrt { 2 }  }  \right)  }^{ 2 } }  } \Rightarrow \quad e=\sqrt { \dfrac { 200 }{ 225 }  } \Rightarrow \dfrac {{2} \sqrt { 2 }  }{ 3 } $$

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