An ellipse <math><mrow><mi>E</mi><mo>:</mo><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math> passes through the vertices of the hyperbola <math><mrow><mi>H</mi><mo>:</mo><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn><mn>9</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>6</mn><mn>4</mn></mrow></mfrac><mo>=</mo><mo>−</mo><mn>1</mn><mo>.</mo></mrow></math> Let the major and minor axes of the ellipse E coincide with transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be <math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>.</mo></mrow></math> If <math><mrow><mi mathvariant="script">l</mi></mrow></math> is the length of the latus rectum of the ellipse E, then the value of 113 <math><mrow><mi mathvariant="script">l</mi></mrow></math> is equal to

eH=1+6449=1137⇒eH.eE=12⇒11349. (64−a2)64=14⇒a2−64=322113l=2a2b=2 (64+322113).18113l=1552

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Conic Sections Ncert Solutions Maths class 11th Maths Class 11th

An ellipse $E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ passes through the vertices of the hyperbola $H : \frac{x^{2}}{49} - \frac{y^{2}}{64} = - 1 .$ Let the major and minor axes of the ellipse E coincide with transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be $\frac{1}{2} .$ If $l$ is the length of the latus rectum of the ellipse E, then the value of 113 $l$ is equal to

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Payal Gupta | Contributor-Level 10

2 months ago

$e_{H} = \sqrt[]{1 + \frac{64}{49}} = \frac{\sqrt[]{113}}{7}$

$\Rightarrow e_{H} . e_{E} = \frac{1}{2}$

$\Rightarrow \frac{113}{49} . \frac{(64 - a^{2})}{64} = \frac{1}{4} \Rightarrow a^{2} - 64 = \frac{3 2^{2}}{113}$

$l = \frac{2 a^{2}}{b} = 2 (64 + \frac{3 2^{2}}{113}) . \frac{1}{8}$

$113 l = 1552$

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