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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

`sqrt2`

D

`sqrt3`

Text Solution

Verified by Experts

The correct Answer is:
B
`(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`
Differentiating w.r.t. x.
`(dy)/(dx)=(db^(2))/(ya^(2))`
Therefore, the slope of normal at (6, 3) is - `a^(2)//2b^(2)`
The equation of normal is
`(y-3)=(-a^(2))/(2b^(2))(x-6)`
It passes through the point (9, 0). Therefore,
`(a^(2))/(2b^(2))=1or(b^(2))/(a^(2))=(1)/(2)." "thereforee^(2)=a+(b^(2))/(a^(2))=1+(1)/(2)`
`therefore" "e=sqrt((3)/(2))`
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Knowledge Check

  • Let P(4,3) be a point on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 . If the normal at P intersects the x-axis at (16,0), then the eccentriclty of the hyperbola is

    A
    `(sqrt(5))/(2)`
    B
    2
    C
    `sqrt(5)`
    D
    `sqrt(3)`

  • If e is the eccentricity of the hyperbola (x^(2))/(a^(2)) - (y^(2))/(b^(2)) = 1 , then e =

    A
    `sqrt(1-(b^(2))/(a^(2)))`
    B
    `sqrt(1+(b^(2))/(a^(2)))`
    C
    `sqrt(1+(a^(2))/(b^(2)))`
    D
    `sqrt(1-(a^(2))/(b^(2)))`

  • The eccentricity of the hyperbola x^2-y^2=4 is

    A
    2
    B
    `2sqrt2`
    C
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