If two tangents to the parabola y 2 = 4ax from a point P make angles θ 1 and θ 2 with the axis of the parabola, then find the locus of P in each of the following cases.
(i) θ 1 + θ 2 = α (a constant)
(ii) θ 1 + θ 2 = 
(iii) tan θ 1 + tan θ 2 = λ (is constant)
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i)y = (x – a) tan α , (ii) x = a, (iii) y = λ x
Sol. Let the equation of tangent and p ≡ (h, k)
k = mh + 
m 2 h – mk + a = 0
m 1 + m 2 = 
...... (i)
m 1 . m 2 = 
..... (ii)
(i) θ 1 + θ 2 = α
tan( θ 1 + θ 2 ) = tan α
tan α = 
⇒ 
= tan α
= tan α
y = (x – a) tan α
(ii) θ 1 + θ 2 = 
⇒ tan θ 1 . tan θ 2 = 1
= 1
x = a
(iii) m 1 + m 2 = λ ⇒ 
= λ
y = λ x
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