Find the measure of the acute angle between the lines represented by:- $x^{2} + y^{2} = 2 x y csc α$ .

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Solution

Equation can be written as $x^{2} + y^{2} - 2 x y csc α = 0$

Comparing it with $a x^{2} + 2 h x y + b y^{2} = 0$
and
$m_{1} + m_{2} = - \frac{2 h}{b} = 2 csc α m_{1} m_{2} = \frac{a}{b} = 1$
$tan θ = ∣ ∣ ∣ \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \frac{\sqrt{(m_{1} + m_{2})^{2} - 4 m_{1} m_{2}}}{1 + m_{1} m_{2}} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \frac{\sqrt{(2 csc α)^{2} - 4 \times 1}}{1 + 1} ∣ ∣ ∣ ∣ = \frac{2 \sqrt{{csc}^{2} α - 1}}{2} = cot α \Rightarrow θ = {tan}^{- 1} cot α = {tan}^{- 1} tan (90^{o} - α) \Rightarrow θ = 90^{o} - α$

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