A straight line is parallel to the lines 3x – y – 3 = 0 and 3x –

1.	A straight line is parallel to the lines 3x – y – 3 = 0 and 3x – y + 5 = 0 and lies between them. Find its equation if its distances from these lines are in the ratio 3 : 5. (a) 3x – y + 10 = 0 (b) 3x – y = 0 (c) 3x – y = 0 (d) 3y – x – 10 = 0
Answer» (b) 3x – y = 0 Given lines are 3x – y – 3 = 0 and 3x – y + 5 = 0. Line parallel to the given lines can be written as 3x – y + c = 0 ...(i) Let us taken a point, say, (0, c) on (i) (Putting x = 0 in (i), we get y = c) ∴ \(\frac{\text{Distance of (0,c) from}\,3x-y-3}{\text{Distance of (0,c) from}\,3x-y+5}\) = \(\frac{3}{5}\) ⇒ \(\frac{\frac{\|3\times0-c-3\|}{\sqrt{3^2+1^2}}}{\frac{\|3\times0-c+5\|}{\sqrt{3^2+1^2}}}\) = \(\frac{3}{5}\)⇒ \(\frac{c+3}{-c+5}\) = \(\frac{3}{5}\) ⇒ 5c + 15 = – 3c + 15 ⇒ 8c = 0 ⇒ c = 0. Substituting c = 0 in (i), the required equation is 3x – y = 0.

A straight line is parallel to the lines 3x – y – 3 = 0 and 3x – y + 5 = 0 and lies between them. Find its equation if its distances from these lines are in the ratio 3 : 5. (a) 3x – y + 10 = 0 (b) 3x – y = 0 (c) 3x – y = 0 (d) 3y – x – 10 = 0

Answer»

(b) 3x – y = 0

Given lines are 3x – y – 3 = 0 and 3x – y + 5 = 0.

Line parallel to the given lines can be written as

3x – y + c = 0 ...(i)

Let us taken a point, say, (0, c) on (i)

(Putting x = 0 in (i), we get y = c)

∴ \(\frac{\text{Distance of (0,c) from}\,3x-y-3}{\text{Distance of (0,c) from}\,3x-y+5}\) = \(\frac{3}{5}\)

⇒ \(\frac{\frac{|3\times0-c-3|}{\sqrt{3^2+1^2}}}{\frac{|3\times0-c+5|}{\sqrt{3^2+1^2}}}\) = \(\frac{3}{5}\)⇒ \(\frac{c+3}{-c+5}\) = \(\frac{3}{5}\)

⇒ 5c + 15 = – 3c + 15 ⇒ 8c = 0 ⇒ c = 0.

Substituting c = 0 in (i), the required equation is 3x – y = 0.

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