Let the hyperbola H : <math><mrow><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> pass through the point <math><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mroot><mrow><mn>2</mn></mrow><mrow></mrow></mroot><mo>,</mo><mo>−</mo><mn>2</mn><mroot><mrow><mn>2</mn></mrow><mrow></mrow></mroot></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?

H : x 2 a 2 − y 2 b 2 = 1 Foci : S (ae, 0), S’ (ae, 0)Focus of parabola is (ae, 0)Now, semi latus rectum of parabola = |SS’| = 2aeGiven, 4ae=e (2b2a)B2 = 2a2……… (i)Given, (22, −22) lies on H⇒1a2−1b2=18........ (ii)From (i) and (ii)a2=4, b2=8? b2=a2 (e2−1)∴e=3⇒ equation of parabola is y2 =83x

Let the hyperbola H :&nbsp;<math><mrow><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>&nbsp;pass through the point&nbsp;<math><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mroot><mrow><mn>2</mn></mrow><mrow></mrow></mroot><mo>,</mo><mo>−</mo><mn>2</mn><mroot><mrow><mn>2</mn></mrow><mrow></mrow></mroot></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math>&nbsp;A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?

Ask & Answer Question

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

Let the hyperbola H : $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ pass through the point $(2 \sqrt[]{2}, - 2 \sqrt[]{2}) .$ A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?

Option 1 -

$(2 \sqrt[]{3}, 3 \sqrt[]{2})$

Option 2 -

$(3 \sqrt[]{3}, - 6 \sqrt[]{2})$

Option 3 -

$(\sqrt[]{3}, - \sqrt[]{6})$

Option 4 -

$(3 \sqrt[]{6}, 6 \sqrt[]{2})$

2 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

2 months ago

Correct Option - 2

Detailed Solution:

$H : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

Foci : S (ae, 0), S’ (ae, 0)

Focus of parabola is (ae, 0)

Now, semi latus rectum of parabola = |SS’| = 2ae

$G i v e n, 4 a e = e (\frac{2 b^{2}}{a})$

B² = 2a²……… (i)

$G i v e n, (2 \sqrt[]{2}, - 2 \sqrt[]{2})$ lies on H

$\Rightarrow \frac{1}{a^{2}} - \frac{1}{b^{2}} = \frac{1}{8} . . . . . . . . (i i)$

From (i) and (ii)

$a^{2} = 4, b^{2} = 8$

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

$∴ e = \sqrt[]{3}$

$\Rightarrow$ equation of parabola is y² $= 8 \sqrt[]{3} x$

0 Upvotes 0 Downvotes

Get authentic answers from experts, students and alumni that you won't find anywhere else

On Shiksha, get access to

65k Colleges
1.2k Exams
688k Reviews
1800k Answers

Community GuidelinesUser Points System

QuestionsDiscussionsTags

1 Answer

Learn more about...

Most viewed information

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question