Question

Given a family of lines a (2x + y + 4) + b(x – 2y – 3) = 0, the number of lines belonging to the family at a distance `sqrt(10)` from P(2, –3) is ______.

Options

• 0

• 1

• 2

• 4

Answer 1

Given a family of lines a (2x + y + 4) + b(x – 2y – 3) = 0, the number of lines belonging to the family at a distance `sqrt(10)` from P(2, –3) is 1.

Explanation:

The length of perpendicular from P(2, –3) on the given family of lines

= `(a(4 - 3 + 4) + b(2 + 6 - 3))/(sqrt((2a + b)^2 + (a - 2b)^2)) = ±sqrt(10)` ...(given)

`\implies` 5a + 5b = `±sqrt(10(5a^2 + 5b^2)`

`\implies` 25(a + b)² = 50(a² + b²)

`\implies` 25(a – b)² = 0

`\implies` a = b

For which we get only line 3x – y + 1 = 0