Question

Consider a curve ax² + 2hxy + by² = 1 and a point P not on the curve. A line drawn from the point P intersects the curve at points Q and R. If the product PQ. PR is independent of the slope of the line, then the curve is

• A. A pair of straight lines
• B. A circle
• C. A parabola
• D. An ellipse or a hyperbola

Answer 1

Answer: (B)

A circle

Answer 2

Let us choose the fixed point P as the origin. Let us now choose any line through P making an angle q with + ve direction of the X- axis (see fig).

Any point on this line can be chosen as

(r cos q, r sin q) where r is the distance measured from P.

If this point must also lie on the given curve, then we have

r²(a cos² q + h sin 2q + h sin 2q + b sin² q) – 1 = 0 ... (i)

For any given q, therefore there will be two values of r.

If PQ and PR be the roots of equation (i), then we have

PQ. PR = – $\frac{1}{f(\theta)}$where f(q) = a cos² q + h sin 2q + b sin² q

Since f(q) is given to be independent of q, therefore using calculus, we have

$\frac{df}{d\theta}$= – 2 a sin q cos q + 2h cos 2q + 2b sin q cos q = 0

i.e. (b – a) sin 2q + 2h cos 2q = 0 ... (ii)

Identically. (The word identically here implies, true for every value of q).

Equation (ii) will be identically true if and only if a = b and

h = 0

which therefore proves that the given curve must be circle.