P(6,4) is a point on the line x−y−2=0. If A(α,β) and B(γ,δ) are two points on this line lying on either side of P at a distance of 4 units from P, then α2+β2+γ2+δ2=
We have a line x−y−2=0 Let x=t is a point on the line (t,t−2) be a parametric point on the line. We have, A and B are two points on either side of P. ∵A=(6+a,4+a)‌ and ‌B(6−a,4−a) Given, PA=4 ‌(6+a−6)2+(4+a−4)2=16 ‌‌‌‌‌begin‌array‌c‌a2+a2=16 a2=8 a=±2√2 ∵‌‌A=(6+2√2,4+2√2)‌ and ‌ B=(6−2√2,4−2√2) A(α,β)‌ and ‌B=(γ,δ) ∵α=6+2√2,β=4+2√2,γ=6−2√2 δ=4−2√2 ∵α2+β2+γ2+δ2 ‌‌=(6+2√2)2+(4+2√2)2+(6−2√2)2+(4−2√2)2 =2(36+8)+2(16+8) =88+48=136