A variable line is drawn through O, to cut two fixed straight lines L1 and L2 in A1 and A2, respectively. A point A is taken on …..

Q: A variable line is drawn through O, to cut two fixed straight lines L1 and L2 in A1 and A2, respectively. A point A is taken on the variable line such that $\displaystyle \frac{m+n}{OA} = \frac{m}{OA_1} + \frac{n}{OA_2} $ . Show that the locus of A is a straight line passing through the point of intersection of L1 and L2 , where O is being the origin.

Sol. Let the variable line passing through the origin is

$\displaystyle \frac{x}{cos\theta} = \frac{y}{sin\theta} = r_i $ ….(i)

Let the equation of the line L1 is p1 x + q1 y = 1 …. (ii)

Equation of the line L2 is p2 x + q2 y = 1 …. (iii)

Let the variable line intersects the line (ii) at A1 and (iii) at A2

Let OA1 = r1

Then A1 = (r1 cosθ , r1 sinθ)

A1 lies on L1

$\displaystyle r_1 = OA_1 = \frac{1}{p_1 cos\theta + q_1 sin\theta}$

$\displaystyle r_2 = OA_2 = \frac{1}{p_2 cos\theta + q_2 sin\theta}$

Let OA = r

Let co-ordinate of A are (h, k)

(h, k) = (r cosθ , r sinθ)

$\displaystyle \frac{m+n}{OA} = \frac{m}{OA_1} + \frac{n}{OA_2} $

$\displaystyle \frac{m+n}{r} = \frac{m}{r_1} + \frac{n}{r_2} $

m + n = m (p1 rcosθ + q1 rsinθ ) + n(p2 rcosθ + q2 rsinθ)

(p1 h + q1 k – 1) + (n/m)(p2 h + q2 k – 1) = 0

Therefore, locus of A is (p1 x + q1 y -1) + (n/m)(p2 x + q2 y -1) = 0

L1 + λ L2 = 0 where λ = n/m

This is the equation of the line passing through the intersection of L1 and L2.