JEE AdvancedParabolaHard

Question

Through the vertex O of the parabola, y2 = 4ax two chords OP and OQ are drawn and the circles on OP and OQ as diameter intersect in R. If θ1, θ2 and f are the angles made with the axis by the tangent at P and Q on the parabola and by OR then the value of cotθ1 + cot θ2 =

Options

A.- 2tan φ
B.- 2 tan(π - φ)
C.0
D.2cot φ

Solution


Let    P(at12, 2at1) & Q(at22, 2at2)
so    tan θ1 = & tan θ2 =
cot θ1 + cot θ2 = t1 + t2    .... (i)
equation of circle with (0, 0) & (at12, 2at1) as end points of diameter is
x(x - at12) + y(y - 2at1) so
S1 : x2 + y2 - at  x - 2at1y = 0    ... (ii)
similarly other circle is
S2 : x2 + y2 - at22 x - 0 at2y = 0        ... (iii)
a(t22  - t12) x + 2a (t2 - t1) y = 0
y = - x
tan φ  = -
from (i)    cot θ1 + cot θ2 = -2 tan φ

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