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Tangent and normal are drawn at P(16,16)...

Tangent and normal are drawn at P(16,16) on the parabola `y^(2)=16x`, which intersect the axis of the parabola at A and B, espectively. If C is the centre of the circle through the points P,A and B and `angleCPB=theta`, then a value of `tantheta` is

A

`(1)/(2)`

B

2

C

3

D

`(4)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
B
Equation of tangent and normal to the curve `y^(2) = 16x` at (16, 16) is `x- 2y + 16 = 0` and `2x + y - 48 = 0` respectively.

`A = (-16, 0) B = (24, 0)`
`:' C` is the centre of circle passing through PAB i.e.,
Slope of `PC = (16 - 0)/(16 - 4) = (16)/(12) = (4)/(3) = m_(1)`
Slope of `PB = (16 - 0)/(16 - 24) = (16)/(-8) = - m_(2)`
`tan theta = |(m_(1) - m_(2))/(1 + m_(1)m_(2))|`
`implies tan theta = |((4)/(3) + 2)/(1 - ((4)/(3))(2))| implies tan theta = 2`
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