Question

If a triangle is formed by any three tangents of the parabola y2 = 4ax two of whose vertices lie on the parabola x2 = 4by then find the locus of the third vertex

Answer 1

Thank you for asking this question, here is your answer:

let y=mx+c  be a tangent

y^2 =4ax.

Now we will solve the above given equations

a=mc.

The Three Tangents will be:

y=ax/c1 +c1

y= ax/c2 +c2

y=ax/c3+c3

We will let the three vertices be

(x12,y12)

(x13,y13)

(x23,y23).

xij = cicj/a and

yij=ci+cj

(x12)^2 = 4by12

and

(x13)^2 =4by13

(c2/c3)^2 = (c1 +c2)/(c1+c3)

It says that c2c3=0

So,  x23=c2c3/a=0 is the locus

and it's a y axis

If there is any confusion please leave a comment below.