Question

The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A'B (where B is the point (2, 3)) subtend angle `π/4` at the origin, is equal to ______.

विकल्प

• 10

• `48/5`

• `52/5`

• 3

Answer 1

The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A'B (where B is the point (2, 3)) subtend angle `π/4` at the origin, is equal to `underlinebb(52/5)`.

Explanation:

Given: A and A' lies on y = 2

Let coordinates of A ≡ (x₁, 2) and A' ≡ (x₂, 2)

Let slope of OA be m₁ and OB be m₂

As we know slope of line passing through (x₁, y₁) and (x₂, y₂) is given by m = `(y_2 - y_1)/(x_2 - x_1)`

⇒ m₁ = `2/x_1` and m₂ = `3/2`

Also, we know that if angle between two lines having slope m₁ and m₂ is θ, then tan θ = `|(m_1 - m_2)/(1 + m_1m_2)|`

⇒ `tan  π/4 = |(2/x_1 - 3/2)/(1 + (2/x_1)(3/2))|`

⇒ 1 = `|(2/x_1 - 3/2)/(1 + 3/x_1)|`

⇒ 1 = `|((4 - 3x_1))/(2(x_1 + 3))|`

⇒ ±1 = `(4 - 3x)/(2(x_1 + 3))`

⇒ 4 – 3x₁ = ±(2x₁ + 6)

⇒ 4 – 3x₁ = 2x₁ + 6

and 4 – 3x = –2x₁ – 6

⇒ 5x₁ = –2 and x₁ = 10

⇒ x₁ = `(-2)/5` and x₁ = 10

∵ x is positive for A and negative for A'

∴ x₁ = 10 and x₂ = `(-2)/5`

⇒ A ≡ (10, 2) and A' ≡ `((-2)/5, 2)`

∴ By distance formula,

AA' = `sqrt((10 + 2/5)^2 + (2 - 2)^2`

⇒ AA' = `52/5`units