analytical geometry problem with locus

Question

A line of constant length 10 units moves with the end always on the -axis and the end always on the line = 4. Find the equation of the locus of the midpoint of . How could I solve this problem?

Answer 1

Let the point on x axis be $A(h,0)$ a general parametric point on the line $y=4x$ is $B(t,4t)$. We can write $$(h-t)^2+4t^2=100~~~(1)$$ Let $P(x,y)$ be the mid point of AB, then $x=(h+t)/2$ and $y=2t$. Puuting $h=2x-t$ and then $t=y/2$ in (1), we get $$(2x-2t)^2+16t^2=100 \implies (2x-y)^2+4y^2=100 \implies 4x^2-4xy+y^2+4y^2=100 \implies 4x^2+5y^2-4xy=100~~~~(2)$$ The Eq. (2) gives the required locus.