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Let P(6, 3) be a point on the hyperbola ...

Let `P(6, 3)` be a point on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1.` If the normal at point P intersects the x-axis at (9, 0), then find the eccentricity of the hyperbola.

A

`sqrt((5)/(2))`

B

`sqrt((3)/(2))`

C

`sqrt(2)`

D

`sqrt(3)`

Text Solution

Verified by Experts

Equation of normal to hyperbola at `(x_(1),y_(1))` is
`(a^(2)x)/(x_(1))+(b^(2)y)/(y_(1))=(a^(2)+b^(2))`
`therefore " At " (6,3)=(a^(2)x)/(6)+(b^(2)y)/(3)=(a^(2)+b^(2))`
` because " It passes through " (9,0). rArr (a^(2)9)/(6)=a^(2)+b^(2)`
`rArr (3a^(2))/(2)-a^(2)=b^(2)rArr (a^(2))/(b^(2))=2`
` therefore e^(2)=1+(b^(2))/(a^(2))=1+(1)/(2) rArr e=sqrt((3)/(2))`
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