Q.67. Hyperbola - Proof when chord of contact is normal.

Question 67.
Tangents are drawn from a point P to the hyperbola x^2 - y^2 = a^2. If the chord of contact of these tangents is normal to the curve, prove that the locus of P is 1/x^2 - 1/y^2 = 4/a^2 .
 
Answer 67.
The equation of the chord of contact to the hyperbola

x^2 - y^2 = a2 at the point P(h, k) is
hx - ky = a^2
Its slope is h/k ... ( 1 )
If it intersects the hyperbola at the point (asecθ, atanθ),
hsecθ - ktanθ = a
=> h - ksinθ = acosθ
=> h - ksinθ = a√(1 - sin^2 θ) ... ( 2 )

Differentiating the equation of the hyperbola,
2x - 2y * dy/dx = 0
=> dy/dx = x/y,
=> slope of the normal to the hyperbola,
- dx/dy = - y/x
=> slope of normal at the point (asecθ, atanθ)
= - atanθ / asecθ = - sinθ
This is the slope of the chord of contact as given in ( 1 )
=> sinθ = - h/k

Plugging this value of sinθ in ( 2 ),
h - k * (- h/k) = a√(1 - h^2/k^2)
=> a√(1 - h^2/k^2) = 2h
=> a^2 (1 - h^2/k^2) = 4h^2
=> (1/h^2) (1 - h^2/k^2) = 4/a^2
=> 1/h^2 - 1/k^2 = 4/a^2
=> P (h, k) lies on the curve,
1/x^2 - 1/y^2 = 4/a^2
which is the locus of P.

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