Question

Show that the foot of the perpendicular from focus to the tangent of the parabola y² = 4ax lies on the tangent at the vertex. 

Answer 1

 Equation of any tangent to the parabola is: 

y=mx+a/m

Q(x₁, y₁) is the foot of the perpendicular 

 ∴ y₁ =mx₁ +a/m    ........ (1)

Slope of SQ= y₁/ x₁-a

SQ and PQ are perpendicular 

Substituting in (1) we get 

 Locus of Q(x₁, y₁) is x = 0 i.e. the tangent at the vertex of the parabola.