Let the eccentricity 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><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><mtext> </mtext><mtext> </mtext><mi>b</mi><mi>e</mi><mtext> </mtext><mtext> </mtext><mroot><mrow><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mrow></mrow></mroot></mrow></math> and length of its latus rectum be <math><mrow><mn>6</mn><mroot><mrow><mn>2</mn></mrow><mrow></mrow></mroot><mo>,</mo><mtext> </mtext><mtext> </mtext><mi>i</mi><mi>f</mi><mtext> </mtext><mtext> </mtext><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math> is a tangent to the hyperbola H, then the value of c2 is equal to

<!- [if gte mso 9]><xml> </xml><! [endif]-> a 2 ( e 2 − 1 ) = b 2 e = 5 2 ⇒ b 2 = 3 a 2 2 <!- [if gte mso 9]><xml> </xml><! [endif]-> Length of latus rectum 2 b 2 a = 6 2 <!- [if gte mso 9]><xml> </xml><! [endif]-> 3 a = 6 2 ⇒ a = 2 2 <!- [if gte mso 9]><xml> </xml><! [endif]-> b = 2 3 <!- [if gte mso 9]><xml> </xml><! [endif]-> y = 2x + c is tangent to hyperbola ∴ c 2 = a 2 m 2 − b 2 = 2 0 <!- [if gte mso 9]><xml> </xml><! [endif]->

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Maths Class 12th Conic Sections Maths NCERT Exemplar Solutions Class 12th Chapter Eleven

Let the eccentricity of the hyperbola $H : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 b e \sqrt[]{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt[]{2}, i f y = 2 x + c$ is a tangent to the hyperbola H, then the value of c²is equal to

Option 1 -

Option 2 -

Option 3 -

Option 4 -

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Payal Gupta | Contributor-Level 10

2 months ago

Correct Option - 2

Detailed Solution:

$a^{2} (e^{2} - 1) = b^{2}$

$e = \sqrt[]{\frac{5}{2}} \Rightarrow b^{2} = \frac{3 a^{2}}{2}$

Length of latus rectum $\frac{2 b^{2}}{a} = 6 \sqrt[]{2}$

$3 a = 6 \sqrt[]{2} \Rightarrow a = 2 \sqrt[]{2}$

$b = 2 \sqrt[]{3}$

y = 2x + c is tangent to hyperbola

$∴ c^{2} = a^{2} m^{2} - b^{2} = 20$

<!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1814000750"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mrow> <mo> (</mo> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math>   <math> <mrow> <mi>e</mi> <mo>=</mo> <mroot> <mrow> <mfrac> <mrow> <mn>5</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> <mo>⇒</mo> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1814000751"> </o:OLEObject></xml><! [endif]-> Length of latus rectum <math> <mrow> <mfrac> <mrow> <mn>2</mn> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>a</mi> </mrow> </mfrac> <mo>=</mo> <mn>6</mn> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1814000752"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mn>3</mn> <mi>a</mi> <mo>=</mo> <mn>6</mn> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> <mo>⇒</mo> <mi>a</mi> <mo>=</mo> <mn>2</mn> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1028" DrawAspect="Content" ObjectID="_1814000753"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1029" DrawAspect="Content" ObjectID="_1814000754"> </o:OLEObject></xml><! [endif]-> y = 2x + c is tangent to hyperbola <math> <mrow> <mo>∴</mo> <msup> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>m</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mn>0</mn> </mrow> </math>        <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1030" DrawAspect="Content" ObjectID="_1814000755"> </o:OLEObject></xml><! [endif]->

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Let the eccentricity of the hyperbola $H : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 b e \sqrt[]{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt[]{2}, i f y = 2 x + c$ is a tangent to the hyperbola H, then the value of c²is equal to

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Let the eccentricity of the hyperbola H:x2a2−y2b2=1 be 52 and length of its latus rectum be 62, if y=2x+c is a tangent to the hyperbola H, then the value of c2 is equal to

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Let the eccentricity of the hyperbola $H : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 b e \sqrt[]{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt[]{2}, i f y = 2 x + c$ is a tangent to the hyperbola H, then the value of c²is equal to