Maths-
General
Easy
Question
From an external point P tangents are drawn to the parabola y2 = 4ax, then the equation to the locus of P when these tangents makes angles θ1 and θ2 with the axis, such that tan θ1 + tan θ2 with the axis, such that tan θ1 + tan θ2 is constant (= b), is -
- y =
- y = bx
- y = b2 x
- None of these
The correct answer is: y = bx
Let the coordinates of P be (h, k) and the equation to the parabola be y2 = 4ax. Any tangent on the parabola is given by y = mx + 
. If this passes through (h, k), the coordinates will satisfy. Hence k = mh + .
m2h – mk + a = 0…(1)
Which is a quadratic in m . Let its roots be m1 and m2, then m1 + m2 = 
and m1m2 = 
. Now, if the two tangents through P make angles θ1 and θ2 with axis of x and m1 = tan θ1 and m2 = tan θ2.
tan θ1 + tanθ2 = .… (2)
and m1m2 = … (3)
tan θ1 tan θ2 = … (4)
By hypothesis tan θ1 + tan θ2 = b
So from equation (2), = b k = bh. Generalising, the locus of (h, k) is y = bx.
Hence (B) is correct answer.
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