NDA 1 2024 Maths, question 58

Topic: Analytical geometry

Q58

If a variable line passes through the point of intersection of the lines $x+2y-1=0$ and $2x-y-1=0$ and meets the coordinate axes in $A$ and $B$ , then what is the locus of the mid-point of $AB$ ?

(a) $3x+y=10xy$
(b) $x+3y=10xy$ Correct answer
(c) $3x+y=10$
(d) $x+3y=10$

Solution.

Answer: (b) $x+3y=10xy$

The fixed intersection of the two lines is (3/5, 1/5).
If (x, y) is the midpoint of the intercepted segment, the axis intercepts are 2x and 2y.
Substitution in intercept form gives (3/5)/(2x) + (1/5)/(2y) = 1, or x + 3y = 10xy.
This is the stated locus equation; the intercept construction uses nonzero finite intercepts.