# Maximum number of common chords that are existent between two Conics

The maximum no. Of common chords between a circle and a parabola is 6. this is because they can have at most Four Points of intersection.

However I have doubt regarding other combination of conics. 1. Parabola and ellipse 2. Parabola and hyperbola 3. Ellipse and circle 4. Ellipse and hyperbola 5. Hyperbola and circle

Also the maximum no. Of common chords b/w 1. Two parabolas 2. Two hyperbolas 3. Two ellipses 4. Two Circles

(I think for two Circles it's 1 and for 2 ellipses it's also 1)

Kindly help me with this problem with a solution explaining the answers.

• It'll be easier for us to help you if you explain some more of your thinking here. What is the nature of your doubt with the various pairings? Have you drawn pictures but not been able to convince yourself that you've gotten the maximum number of intersections? Could you show us those pictures? (In the case of two ellipses, have you considered that they might not be oriented in the same way?) – Blue Apr 28 at 7:15
• I have doubt regarding how many common chords can any pairing of the above Conics have. I could solve for Parabola and circle but the rest I am having problem. Also the thing about the orientation also confuses me as we have to take that in consideration and that complicates things a bit. – Arg10 Apr 28 at 7:18
• Do you mean that you "solved" by using equations and getting four points of intersection? Can you show your work, and try to convey why you're having difficulty applying the technique to other cases? The equations can get a little tricky, but I don't want to waste time (yours or mine) telling you things you already know. Even so, you should draw a few pictures to build your intuition about what to expect. (With the ellipses, consider the case where one is short-and-wide and the other is tall-and-narrow.) – Blue Apr 28 at 7:27
• Ok Thanks, I will try it again – Arg10 Apr 28 at 7:29