For the parabola y²=4x, AB and CD are any two parallel chords having slope 1. C1 is a circle passing through O, A and B and C2 is a circle passing through O, C and D, where O is origin. C1 and C2 Intersect at -?
I know that is A(t1) and B(t2) are points on the parabola having slope 1 then t1+t2=2, but i did not get what property is used in the solution where they simply add t1+t2+t5=0 to find the answer. Please view the attached image of the solution.
Let $x^2+y^2+2gx+2fy+c=0$ be the equation of a generic circle. Plugging there $x=t^2,\ y=2t$ we get the equation for the intersections between circle and parabola: $$ t^4+(4+2g)t^2+4ft+c=0. $$ In the case of circle $C_1$ we know this equation has three real solutions $(t_1,t_2,0)$, hence it also has a fourth real solution, let's call it $t_5$. The sum of these solutions is the opposite of the coefficient of $t^3$ in the equation, which in our case is vanishing. Hence: $$ t_1+t_2+0+t_5=0 $$ and an analogous reasoning can be done for circle $C_2$.