- #1

utkarshakash

Gold Member

- 855

- 13

## Homework Statement

Locus of the point of intersection of tangents to the parabolas

*y*[itex]^{2}[/itex]=4(

*x*+1) and

*y*[itex]^{2}[/itex]=8(

*x*+2) which are at right angles, is

## Homework Equations

Equation of tangent for first parabola

t[itex]_{1}[/itex]

*y*=

*x*+1+at[itex]_{1}[/itex][itex]^{2}[/itex]

Equation of tangent for second parabola

t[itex]_{2}[/itex]

*y*=

*x*+2+bt[itex]_{2}[/itex][itex]^{2}[/itex]

## The Attempt at a Solution

Let us assume that the point of intersection of tangents is (

*h,k*)

Since it lies on the tangent

∴kt[itex]_{1}[/itex]=h+1+at[itex]_{1}[/itex][itex]^{2}[/itex]

[itex]\Rightarrow[/itex]at[itex]_{1}[/itex][itex]^{2}[/itex]-kt[itex]_{1}[/itex]+(h+1)=0

t[itex]_{1}[/itex]t[itex]_{2}[/itex]=h+1 (product of roots)

Also t[itex]_{1}[/itex]t[itex]_{2}[/itex]=-1 (Since they are at right angles)

∴required locus = x+2=0

I can't understand where I'm wrong. Is the equation of tangent incorrect? Please Help.