Question

Prove that the equation to the straight lines through the origin, each of which makes an angle α with the straight line y = x is x² – 2xy sec 2α + y² = 0

Answer 1

Slope of y = x is m = tan θ = 1 

⇒ θ = 45° 

The new lines slopes will be 

m = tan(45 + α) and m = tan (45 – α) 

∴ The equations of the lines passing through the origin is given by 

y = tan(45 + α)x and y = tan(45 – α)x 

(i.e) y = tan(45 + α)x = 0 and y = tan(45 – α)x = 0

The combined equation is [y – tan (45 + α)x] [y – tan (45 – α)x] = 0 

y² + tan(45 + α)tan(45 – α)x² – xy[tan(45 – α) + tan(45 + α)] = 0

Let the equation of lines passes through the origin 

So the equations are y = m₁ x = 0 and y = m₂ x = 0 

So the combined equations is (y – m₁ x) (y – m₂ x) = 0 

(i.e) y² – xy(m₁ + m₂) + m₁ m₂ x = 0 

(i.e) y² – xy(2sec α) + x² (1) = 0 

(i.e) y² – 2xy sec 2α + x² = 0