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If two tangents to the parabola y 2 =4ax from a point P make an angle θ 1 and θ 2 with the axis of the parabola, then find the locus of P if cosθ 1 cosθ 2 =λ (a constant)

If two tangents to the parabola y2=4ax from a point P make an angle θ1 and θ2 with the axis of the parabola, then find the locus of P if cosθ1cosθ2=λ (a constant)

Grade:upto college level

1 Answers

Pratik Tibrewal
askIITians Faculty 37 Points
7 years ago
Let the equation of tangent in slope form be y= mx + a/m
Let point P be (h,k)
hence this point will satisfy the line
there fore: k = mh + a/m or mh^2 - km + a = 0 (a quadratic in 'm' which has two roots m1 (tan(a)and m2 tan(b))
gicen cos(a) . cos(b) = K;
from quadratic product of roots: tan(a) . tan(b) = a/h^2 -------- (i)
sum of roots : tan(a) + tan(b) = k/h^2
now sec^2(a). sec^2(b) =(1+tan^2(a))(1+tan^2(b)) = 1 + tan^2(a) + tan^2(b) + tan^2(a) . tan^2(b) =
= 1 + (tan(a) + tan(b))^2 - 2tan(a) . tan(b) + tan^2(a). tan^2(b)
= 1 + k^2/h^4 - 2a/h^2 + a^2/h^4
now cos (a) . cos(b) = 1/sec(a). sec(b) = K


Thanks and Regards,

Pratik Tibrewal
askiitian faculty,
BTech, IITG


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