Tangents are drawn to the parabola `y^2=4a x` at the point where the line `l x+m y+n=0` meets this parabola. Find the point of intersection of these t

Question

Tangents are drawn to the parabola `y^2=4a x` at the point where the line `l x+m y+n=0` meets this parabola. Find the point of intersection of these t

Tangents are drawn to the parabola `y^2=4a x` at the point where the line `l x+m y+n=0` meets this parabola. Find the point of intersection of these tangents.

Monjur Alahi

5 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Given line is
`lx+my+n=0`
Let this intersect parabola `y^(2)=4ax` at points P and Q.
Now, tangents drawn to parabola at points P and Q, intersect at R(h,k). lt,brgt So, PQ is chord off contact of parabola w.r.t. point R(h,k).
so, equation of line PQ is
`ky=2a(x+h)`
`or2ax-ky+2ah=0` (2)
Equation (1) and (2) represent the same straight line
`:." "(2a)/(l),=(-k)/(m)=(2ah)/(m)`
`rArr" "h=(n)/(l),k=-(2am)/(l)`
So, required point of intersection is `((n)/(l),-(2am)/(l))`.