Let L be a common tangent line to the curves $4 x^{2} + 9 y^{2} = 36 a n d {(2 x)}^{2} + {(2 y)}^{2} = 31 .$ Then the square of the slope of the line L is………….

2 Views | Posted 2 weeks ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 weeks ago

Given curves $\frac{x^{2}}{9} + \frac{y^{4}}{4} = 1 . . . . . . . . . (i)$

& $x^{2} + y^{2} = \frac{31}{4} . . . . . . . . . . . (i i)$

Equation of any tangent to (i) be y = mx + $\sqrt[]{9 m^{2} + 4} . . . . . . . . . . . . (i i i)$

For common tangent (iii) also should be tangent to (ii) so by condition of common tangency

$9 m^{2} + 4 = \frac{31}{4} (1 + m^{2})$

OR 36m² + 16 = 31 + 31m²

->m² = 3

Given curves <math> <mrow> <mfrac> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>9</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mrow> <mi>y</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo> (</mo> <mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>  <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1819617314"> </o:OLEObject></xml><! [endif]->&       <math> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mn>1</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo> (</mo> <mrow> <mi>i</mi> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>   <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1819617315"> </o:OLEObject></xml><! [endif]->Equation of any tangent to (i) be y = mx + <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1819617316"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mroot> <mrow> <mn>9</mn> <msup> <mrow> <mi>m</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> </mrow> <mrow></mrow> </mroot> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo> (</mo> <mrow> <mi>i</mi> <mi>i</mi> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> For common tangent (iii) also should be tangent to (ii) so by condition of common tangency <math> <mrow> <mn>9</mn> <msup> <mrow> <mi>m</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mn>1</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mrow> <mo> (</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mi>m</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </math>          <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1028" DrawAspect="Content" ObjectID="_1819617317"> </o:OLEObject></xml><! [endif]->OR 36m2 + 16 = 31 + 31m2->m2 = 3

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

Let L be a common tangent line to the curves $4 x^{2} + 9 y^{2} = 36 a n d {(2 x)}^{2} + {(2 y)}^{2} = 31 .$ Then the square of the slope of the line L is………….

1 Answer

Learn more about...

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

Let L be a common tangent line to the curves 4 x 2 + 9 y 2 = 3 6 a n d ( 2 x ) 2 + ( 2 y ) 2 = 3 1 . Then the square of the slope of the line L is………….

1 Answer

Learn more about...

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

Let L be a common tangent line to the curves $4 x^{2} + 9 y^{2} = 36 a n d {(2 x)}^{2} + {(2 y)}^{2} = 31 .$ Then the square of the slope of the line L is………….