Question

Find the distance between the points a(at 12 2at1) b(at22 2at2)

Answer 1

Answer:

Distance between two points (a,b) and (p,q) is

   $=\sqrt{(a-p)^2+(b-q)^2$

Now, we have to find distance between points , $(at_{1}^2, 2a t_{1}),and (at_{2}^2, 2a t_{2})=\sqrt{((at_{1}^2-at_{2}^2)^2+( 2a t_{1} - 2a t_{2})^2}\\\\ =\sqrt{a^2(t_{1}^2-t_{2}^2)^2+4a^2(t_{1}-t_{2})^2}\\\\ =\sqrt{a^2(t_{1}^4+t_{2}^4-2t_{1}^2t_{2}^2+4t_{1}^2+4t_{2}^2-8t_{1}t_{2}})\\\\ =a\sqrt{t_{1}^4+t_{2}^4-2t_{1}^2t_{2}^2+4t_{1}^2+4t_{2}^2-8t_{1}t_{2}$