A line passes through the point of intersection of the lines 100x + 50

1.	A line passes through the point of intersection of the lines 100x + 50y – 1 = 0 and 75x + 25y + 3 = 0 and makes equal intercepts on the axes. Its equation is(a) 25x – 25y + 6 = 0 (b) 5x – 5y + 3 = 0 (c) 25x + 25y – 4 = 0 (d) 5x + 5y – 7 = 0
Answer» (c) 25x + 25y – 4 = 0 The point of intersection of the given lines can be obtained by solving the equations of the two lines simultaneously. 100x + 50y = 1 ...(i) 75x + 25y = –3 ...(ii) Eqn (i) – 2 × Eqn (ii) ⇒ (100x + 50y) – (150x + 50y) = 1 – (–6) ⇒ – 50x = 7 ⇒ x = \(\frac{-7}{50}\) 100 x \(\frac{-7}{50}\) + 50y = 1 ⇒ 50y = 1 + 14 = 15 ⇒ y = \(\frac{15}{50}\) = \(\frac{3}{10}\) Let the required line make x-intercept = y-intercept = a. Then eqn of required line is \(\frac{x}{a}\) + \(\frac{y}{a}\) = 1 ⇒ x + y = a Since it passes through \(\bigg(\)\(\frac{-7}{50}\), \(\frac{3}{10}\)\(\bigg)\), therefore ⇒ \(\frac{-7}{50}\) + \(\frac{3}{10}\) = a ⇒ \(\frac{-7+15}{50}\) = a ⇒ a = \(\frac{8}{50}\) = \(\frac{4}{25}\) ∴ Eqn of required line: x + y = \(\frac{4}{25}\) ⇒ 25 + 25y – 4 = 0.

A line passes through the point of intersection of the lines 100x + 50y – 1 = 0 and 75x + 25y + 3 = 0 and makes equal intercepts on the axes. Its equation is(a) 25x – 25y + 6 = 0 (b) 5x – 5y + 3 = 0 (c) 25x + 25y – 4 = 0 (d) 5x + 5y – 7 = 0

Answer»

(c) 25x + 25y – 4 = 0

The point of intersection of the given lines can be obtained by solving the equations of the two lines simultaneously.

100x + 50y = 1 ...(i)

75x + 25y = –3 ...(ii)

Eqn (i) – 2 × Eqn (ii)

⇒ (100x + 50y) – (150x + 50y) = 1 – (–6)

⇒ – 50x = 7 ⇒ x = \(\frac{-7}{50}\)

100 x \(\frac{-7}{50}\) + 50y = 1 ⇒ 50y = 1 + 14 = 15 ⇒ y = \(\frac{15}{50}\) = \(\frac{3}{10}\)

Let the required line make x-intercept = y-intercept = a.

Then eqn of required line is \(\frac{x}{a}\) + \(\frac{y}{a}\) = 1 ⇒ x + y = a

Since it passes through \(\bigg(\)\(\frac{-7}{50}\), \(\frac{3}{10}\)\(\bigg)\), therefore

⇒ \(\frac{-7}{50}\) + \(\frac{3}{10}\) = a ⇒ \(\frac{-7+15}{50}\) = a ⇒ a = \(\frac{8}{50}\) = \(\frac{4}{25}\)

∴ Eqn of required line: x + y = \(\frac{4}{25}\) ⇒ 25 + 25y – 4 = 0.