Answer to Question #223989 in Analytic Geometry for Bless

Question

The tangent to the parabola y²= 4ax at the point P(at² , 2at) meets the x-axis at A and the y-axis at B. Find the equation of the locus of the mid-point of AB at t varies.

Answer 1

Solution;

Let;

x₁=at² and y₁=2at

Such that;

At (x₁,y₁),we have the tangent equation as;

yy₁=2a(x+x₁)

By substitution;

2aty=2a(x+at²)

ty=x+ at² is the tangent at point P.

Now,using the equation of the tangent,we find the intersection points A and B.

At A, y=0

ty=0+at²

y=at

At B,x=0

t(0)=x+at²

x=-at²

Take the midpoint point of A and B as C{h,k}

Such that;

$h=\frac{0+-at^2}{2}$

$h=\frac{-at^2}2$

-2h=at²

$k=\frac{0+at}{2}$

$k=\frac{at}2$

2k=at

Since ;

y²=4ax,the equation of parabola will be ;

(2at)²=4a(at²)

From h and k;

(2×2k)²=4a(-2h)

16k²=-8ah

2k²=-ah

Rewrite as;

2k²+ah=0

Hence,

2y²+ax=0