If the line x-2y=12 is the tangent to the ellipse

 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 at the point (3,\frac{-9}{2}) , then the length

of the latus rectum of the ellipse is : 

  • Option 1)

    9

  • Option 2)

    12\sqrt2

  • Option 3)

    5

  • Option 4)

    8\sqrt3

 

Answers (1)

Tangent to a given ellipse at (x_1,y_1)

\frac{xx_1}{a^{2}}+\frac{yy_1}{b^{2}}=1

Equation of tangent at (3,\frac{-9}{2})

\frac{3x}{a^{2}}-\frac{9y}{2b^{2}}=1

Now compare this equation with given equation of tangent  x - 2y = 12

\frac{3}{a^{2}}=\frac{9}{4b^{2}}=\frac{1}{12}

a=6\: \: and\: \: b=3\sqrt3

Length of LR = \frac{2b^{2}}{a}=\frac{2\times(3\sqrt3)^{2} }{6}=9

So, correct  option is (1).


Option 1)

9

Option 2)

12\sqrt2

Option 3)

5

Option 4)

8\sqrt3

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