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………….

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alok kumar singh | Contributor-Level 10

a month 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

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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………….

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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………….

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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………….