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if tangents to the parabola y^2=4ax intersect the hyperbola x^2/a^2 - y^2/b^2 = 1 at A and B , then find the locus of point of intersection of tangents at A and B.

if tangents to the parabola y^2=4ax intersect the hyperbola x^2/a^2 - y^2/b^2 = 1 at A and B , then find the locus of point of intersection of tangents at A and B.

Grade:11

1 Answers

Samyak Jain
333 Points
2 years ago
Equation of parabola is y2 = 4ax. 
Equation of hyperbola is x2 / a2 – y2 / b2 = 1.
Let Q be a point on the parabola at which tangent is drawn which intersects the hyperbola at A and B.
Let tangents drawn at A and B of the hyperbola meet at P(h,k).
\therefore  AB is the chord of contact of hyperbola from external point P.
Equation of AB is T=0 wrt to hyperbola which is 
xh/a2 – yk/b2 = 1                          …..(1)
Also line AB is tangent to the parabola at Q. Let its slope be m.
\therefore  Equation of AB is y = mx + a/m
i.e. mx – y = – a/m                       …..(2)
Comparing (1) & (2), we get
[(h/a2)/m] = [k/b2] = [1/(– a/m)]
\Rightarrow h/ma2 = k/b= – m/a
m = b2h/a2k    &    m = – ak/b2
\therefore b2h/a2k = – ak/b2   or    b4h = – a3k2
Replace h by x and k by y to get locus of P.
b4x = – a3y2
i.e. b4x + a3y =  0 is the locus of point of intersection of tangents at A and B.

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