Need help in 2D coordinate geometry problem of pair of straight lines?

If the lines have slopes m₁ & m₂ their combined equation is (y−m₁x)(y−m₂x) = 0

Expand this to y²−(m₁+m₂)xy+m₁m₂x² = 0 so you just need to find m₁+m₂ & m₁m₂ … (i)

For a line y=mx the distance of (p.q) from it is |q−mp|/√(1+m²)

If this is k then (q−mp)²/(1+m²) = k² which simplifies to m²(k²−p²)+2pqm+(k²−q²) = 0

By Vieta, if the solutions of this quadratic in m are m₁, m₂ then

m₁+m₂ = −2pq/(k²−p²) and m₁m₂ = (k²−q²)/(k²−p²)

Sub these in (i) to get y² +2pqxy/(k²−p²) + (k²−q²)x²/(k²−p²) = 0

⟹ (k²−p²)y² +2pqxy + (k²−q²)x² = 0

⟹ k²(x²+y²) = p²y²−2pqxy+q²x²

⟹ k²(x²+y²) = (py−qx)²