From an external point P, tangents are drawn to the parabola; find the equation of the locus of P when these tangents make angles θ 1 and θ 2 with the axis, such that cos θ 1
cos θ 2 = µ , which is constant.
Text Solution
Verified by Experts(iii) (i); (iv) (ii); (vi) (iii)
x 2 = µ 2 {(x – a) 2 + y 2 }
Sol.
y = mx + 
k = mh + 
m 2 h – mk + a = 0
m 1 = tan θ 1
m 2 = tan θ 2
m 1 + m 2 = 
...... (i)
m 1 m 2 = 
.... (ii)
from (i) tan θ 1 + tan θ 2 = 
....... (iii) (i) tan θ 1 + tan θ 2 = 
....... (iii)
from (ii) tan θ 1 . tan θ 2 = 
........ (iv) (ii) tan θ 1 . tan θ 2 = 
........ (iv)
cos θ 1 · cos θ 2 = μ (given) ........ (v)
from (iii) sin( θ 1 + θ 2 ) = 
....... (vi) (iii) sin( θ 1 + θ 2 ) = 
....... (vi)
from (iv) sin θ 1 · sin θ 2 = 
(iv) sin θ 1 · sin θ 2 = 
∴ cos( θ 1 + θ 2 ) = μ – 
...... (vii)
squaring and adding (vi) & (vii) we get the required locus 
⇒ 
⇒ μ 2 [y 2 + (x – a) 2 ] = x 2 .
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