How to find the set of values of a for which the line $y+x=0$ bisects the two chords from the point $P(\frac{1+\sqrt{2a}}{2},\frac{1-\sqrt{2a}}{2})$ to the circle $2x^2+2y^2-(1+\sqrt{2a})x-(1-\sqrt{2a})y=0$ ?
Got stuck with this problem.Cant understand what does "two chords" mean here?And btw what should be the right approach?
The following is my interpretation of your problem. Hope it helps.
Let $1 + \sqrt {2a} = t$. Then, $1 – \sqrt {2a} = t’$, the corresponding surd conjugate.
Observations:-
(1) x + y = 0 is a line through O, the origin.
(2) Circle C (with center C) passes through O.
(3) P is on circle C.
(4) PC is a line through O.
Thus, we have the following diagram with
(1.1) PMH and PNK are the bisected chords.
(1.2) PM = MK and PN = NK.
(1.3) $\angle CMH = \angle CNk = 90^0$