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Question 1209672: An ellipse and a hyperbola have the same foci, $A$ and $B$, and intersect at four points. The ellipse has major axis $24,$ and minor axis $13.$ The distance between the vertices of the hyperbola is $5$. Let $P$ be one of the points of intersection of the ellipse and hyperbola. What is $PA \cdot PB$?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
1. **Ellipse Properties:**
* The major axis of the ellipse is 2a = 24, so a = 12.
* The minor axis of the ellipse is 2b = 13, so b = 13/2 = 6.5.
* The distance from the center to each focus (c) is related to a and b by c² = a² - b².
* c² = 12² - (6.5)² = 144 - 42.25 = 101.75
* c = √101.75
2. **Hyperbola Properties:**
* The distance between the vertices of the hyperbola is 2a' = 5, so a' = 2.5. (We use a' to distinguish it from the ellipse's semi-major axis).
* The foci of the hyperbola are the same as the ellipse, so the distance from the center to each focus is also c = √101.75.
* For a hyperbola, c² = a'² + b'², where b' is related to the hyperbola.
* 101.75 = (2.5)² + b'²
* b'² = 101.75 - 6.25 = 95.5
* b' = √95.5
3. **Intersection Point P:**
* For any point on the ellipse, the sum of the distances to the foci is constant and equal to the major axis: PA + PB = 2a = 24.
* For any point on the hyperbola, the *difference* of the distances to the foci is constant and equal to the distance between the vertices: |PA - PB| = 2a' = 5.
4. **Solving for PA and PB:**
We have two equations:
* PA + PB = 24
* |PA - PB| = 5
Case 1: PA - PB = 5
Adding the two equations: 2PA = 29 => PA = 14.5
PB = 24 - 14.5 = 9.5
PA * PB = 14.5 * 9.5 = 137.75
Case 2: PB - PA = 5
Adding the two equations: 2PB = 29 => PB = 14.5
PA = 24 - 14.5 = 9.5
PA * PB = 14.5 * 9.5 = 137.75
In either case, PA * PB = 137.75.
Therefore, PA * PB = 137.75.
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