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if the line x- 2y = 12 is tangent ...

if the line x- 2y = 12 is tangent to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` at the point `(3,(-9)/(2))` then the length of the latusrectum of the ellipse is

A

`8 sqrt(3)`

B

`9`

C

5

D

`12 sqrt(2)`

Text Solution

Verified by Experts

The correct Answer is:
B
KEy idea write equation of the tangent to the ellipse point and use formula for latusrectum of ellipse .
Equation of given ellipse is
`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` .
Now , equation of tangent at the point `(3,-(9)/(2))` on the ellipse(i) is
`implies (3x)/(a^(2))=(9y)/(2b^(2))=1`
`[ :' ` the equation of the tangent to the ellipse` (x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
at the point `(x_(1), y_(1)) ` is `(xx_(1))/(a^(2))+(yy_(1))/(b^(2))=1]`
`:'` Tangent (ii) represent the line `x-2y= 12 ,` so
`(1)/(3)=(2)/(9)=(12)/(1)`
`a^(2) 2b^(2)`
`implies a^(2)=36 and b^(2)=27`
Now , Length of latusrectum `=(2b^(2))/(a)=(2xx27)/(6)=9` units
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