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The tangents and normal at a point on (x...

The tangents and normal at a point on `(x^(2))/(a^(2))-(y^(2))/(b^(2)) =1` cut the y-axis A and B. Then the circle on AB as diameter passes through

A

a)one of the vertex of the hyperbola

B

b)one of the foot of directrix on x-axis of the hyperbola

C

c)foci of the hyperbola

D

d)none of these

Text Solution

Verified by Experts

The correct Answer is:
C
Equation of tangent at point `P(theta)` is
`(sec theta)/(a) x -(tan theta)/(b) y =1`
`:. A (0,-b cot theta)`
Equation of normal at point `P(theta)` is
`a cos theta x + b cot theta y = a^(2) +b^(2)`
`:. B(0,(a^(2)+b^(2))/(b cot theta))`
Equation of circle as AB as a diameter is
`(x-0) (x-0) + (y+b cot theta) (y-(a^(2)+b^(2))/(b cot theta)) =0`
or `x^(2)+ (y+b cot theta) (y-(a^(2)e^(2))/(b cot theta)) =0`
Clearly this passes through foci (ae,0)
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