Question

Find 'k' if the sum of slopes of lines represented by equation x² + kxy − 3y² = 0 is twice their product.

Answer 1

Comparing the equation x² + kxy – 3y² = 0 with ax² + 2hxy + by² = 0, we get, a = 1, 2h = k, b = −3. 

Let m₁ and m₂ be the slopes of the lines

represented by x² + kxy − 3y² = 0.

∴ m₁ + m₂ = ${\displaystyle \frac{-2h}{b}=-\frac{k}{(-3)}=\frac{k}{3}}$

and m₁m₂ = ${\displaystyle \frac{a}{b}=\frac{1}{(-3)}=-\frac{1}{3}}$

Now, m₁ + m₂ = 2(m₁m₂) ..(Given)

∴ ${\displaystyle \frac{k}{3}=2(-\frac{1}{3})}$

∴ k = −2