1. ## Conics help

I need help with part 3 of this question

The ellipse E1 has equation x^2/a^2 + y^2/b^2=1 and ellipse E2 has equation x^2/b^2 + y^2/a^2=1. The point P(x0,y0) lies outside both E1 and E2.

i) Find all the points of intersection of E1 and E2

ii) The chord of contact of E1 from the point P(x0,y0) has equation xx0/a^2 + yy0/b^2 =1. This chord of contact intersects with the chord of contact to E2 from the point P(x0,y0) at the point Q(x2,y2). Find the coordinates of Q

iii) Using your answer to part (i) or otherwise show that Q cannot lie outside both E1 and E2

2. ## Re: Conics help

Is it actually true that $Q$ must always be inside at least one of the ellipses?

Suppose $P$ lies on either of the co-ordinate axes. Then the chords of contact from $P$ to both ellipses would be parallel (since the ellipses are 90 degree rotations of each other) and there would be no intersection.

Now suppose you moved $P$ a tiny amount such that it no longer lies on the co-ordinate axis. Then the chords of contact should be very slightly slanted to each other and their intersection point $Q$ should not be anywhere near the ellipses.

So unless I read the question wrong, it seems like the last part is asking you to prove something that isn't true.

3. ## Re: Conics help

Yeah I also think the question is not true, counter examples can be illustrated here:
https://www.desmos.com/calculator/fhj9angyd2

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