A straight line passes through the points (a, 0) and (0, b). The lengt

1.	A straight line passes through the points (a, 0) and (0, b). The length of the line segment contained between the axes is 13 and the product of the intercepts is 60. Find the equation of the straight line (a) 5x + 12y = 60 (b) 7x – 12y = 50 (c) 5x + 12y + 60 = 0 (d) Both (a) and (c)
Answer» (d) Both (a) and (c) Since the line passes through A(a, 0) and B(0, b), it makes intercepts a and b on x-axis and y-axis respectively. Let the equation of this line in the intercept from be \(\frac{x}{a}\) + \(\frac{y}{a}\) = 1 By the given condition, AB = \(\sqrt{a^2+b^2}\) = 13 and ab = 60 a² + b² = 169 ⇒ a² + b² + 2ab = 169 + 120 ⇒ (a + b)² = 289 ⇒ a + b = ±17 ...(i) Also, a² + b² – 2ab = 169 – 120 ⇒ (a – b)² = 49 ⇒ a – b = ±7. ...(ii) ∴ From (i) and (ii) a = 12, b = 5 and a = –12, b = –5 ∴ The required equations of the straight line are: \(\frac{x}{12}\) + \(\frac{y}{5}\) = 1 and \(\frac{x}{-12}\) + \(\frac{y}{-5}\) = 1 ⇒ 5x + 12y = 60 and 5x + 12y + 60 = 0.

A straight line passes through the points (a, 0) and (0, b). The length of the line segment contained between the axes is 13 and the product of the intercepts is 60. Find the equation of the straight line (a) 5x + 12y = 60 (b) 7x – 12y = 50 (c) 5x + 12y + 60 = 0 (d) Both (a) and (c)

Answer»

(d) Both (a) and (c)

Since the line passes through A(a, 0) and B(0, b), it makes intercepts a and b on x-axis and y-axis respectively.

Let the equation of this line in the intercept from be \(\frac{x}{a}\) + \(\frac{y}{a}\) = 1

By the given condition, AB = \(\sqrt{a^2+b^2}\) = 13 and ab = 60

a² + b² = 169 ⇒ a² + b² + 2ab = 169 + 120

⇒ (a + b)² = 289 ⇒ a + b = ±17 ...(i)

Also, a² + b² – 2ab = 169 – 120 ⇒ (a – b)² = 49

⇒ a – b = ±7. ...(ii)

∴ From (i) and (ii) a = 12, b = 5 and a = –12, b = –5

∴ The required equations of the straight line are:

\(\frac{x}{12}\) + \(\frac{y}{5}\) = 1 and \(\frac{x}{-12}\) + \(\frac{y}{-5}\) = 1

⇒ 5x + 12y = 60 and 5x + 12y + 60 = 0.