Asked by nini
Please help with this problem:
An ellipse and a hyperbola have the same foci, $A$ and $B$, and intersect at four points. The ellipse has major axis 50, and minor axis 40. The hyperbola has conjugate axis of length 20. Let $P$ be a point on both the hyperbola and ellipse. What is $PA \cdot PB$?
An ellipse and a hyperbola have the same foci, $A$ and $B$, and intersect at four points. The ellipse has major axis 50, and minor axis 40. The hyperbola has conjugate axis of length 20. Let $P$ be a point on both the hyperbola and ellipse. What is $PA \cdot PB$?
Answers
Answered by
Steve
If you set up the curves as described, you have
x^2/25^2 + y^2/20^2 = 1
x^2/125 - y^2/100 = 1
They intersect at (±25√5/3,±40/3) so pick P to be one of those points and figure your answer.
x^2/25^2 + y^2/20^2 = 1
x^2/125 - y^2/100 = 1
They intersect at (±25√5/3,±40/3) so pick P to be one of those points and figure your answer.
Answered by
Sammy
Steve means the intersections are (±(25√5)/3,±(40)/3) if anyone is confused.
Focus 1 (A): (15, 0)
Focus 2 (B): (-15, 0)
P: ((25√5)/3,(40)/3)
The question is asking for the distance between P and A, and multiply that by the distance between P and B.
Distance between P and A: √(750 - 250√5)
Distance between P and B: √(750 + 250√5)
Our final answer is 500, which is correct.
Focus 1 (A): (15, 0)
Focus 2 (B): (-15, 0)
P: ((25√5)/3,(40)/3)
The question is asking for the distance between P and A, and multiply that by the distance between P and B.
Distance between P and A: √(750 - 250√5)
Distance between P and B: √(750 + 250√5)
Our final answer is 500, which is correct.
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.