Question

Difference of slopes of the lines represented by equation x²(sec²θ - sin²θ) - 2xy tanθ + y² sin²θ = 0 is ______ 

Options

• 1 

• 3

• 2

• 4

Answer 1

Difference of slopes of the lines represented by equation x²(sec²θ - sin²θ) - 2xy tanθ + y² sin²θ = 0 is 2.

Explanation:

Given equation of pair of lines is

x²(sec²θ - sin²θ) - 2xy tanθ + y² sin²θ = 0

∴ a = sec²θ - sin²θ, h = -tanθ, b = sin²θ

Now, `m_1 + m_2 = (2tantheta)/(sin^2theta)`

`m_1m_2 = (sec^2theta - sin^2theta)/(sin^2theta)`

∴ `m_1 - m_2 = sqrt((m_1 + m_2)^2 - 4m_1m_2)`

= `sqrt(((2tantheta)/sin^2theta)^2 - 4((sec^2theta - sin^2theta)/(sin^2theta)))`

= `sqrt((4tan^2theta)/(sin^4theta) - 4(sec^2theta cosec^2theta - 1))`

= `sqrt(4sec^2theta cosec^2theta - 4sec^2theta cosec^2theta + 4)`

= 2