Q.150. Coordinate geometry - Lines

Question 150.
A (3, 0) and B(6, 0) are two constant points and U(h, k) is a variable point of the plane. AU and BU meet the oordinate axis(y-axis) at C and D respectively and AD meets OU at V. Prove that CV passes through (2, 0) for any position of U in the plane.

Ans. 150.

Let C = (0, c) and D = (0, d)
=> eqn. of line AC is x/3 + y/c = 1 and
eqn. of line BD is x/6 + y/d = 1
As both pass through (h, k),
h/3 + k/c = 1 and h/6 + k/d = 1
Subtracting, (h/3 - h/6) = k/d - k/c ... ( 1 )

As O, V, U are collinear, let V = (rh, rk)
Let CV intersect x-axis in P(p, 0)
Eqn. of CV is x/p + y/c = 1 and
eqn. of AD is x/3 + y/d = 1
As V (rh, rk) lies on AD and CV,
rh/p + rk/c = 1 and rh/3 + rk/d = 1
=> h/p + k/c = 1/r and h/3 + k/d = 1/r
Subtracting, h/p - h/3 = k/d - k/c ... ( 2 )

From ( 1 ) and ( 2 ),
h/p - h/3 = h/3 - h/6
=> 1/p = 2/3 - 1/6 = 1/2 => p = 2.
=> CV intersects x-axis at P(2, 0)
for all h ∈ R - {3, 6} and all k ∈ R.

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