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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?

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Khoso123

Answer:

PA.PB=600

Step-by-step explanation:

Given Data

Major axis=50

Minor axis=40

conjugate axis of length=20

P be a point on both the hyperbola and ellipse

PA.PB=?

Solution

Taken definition of ellipse

PA+PB=50

Minor axis of ellipse is 40 then a=25 and b=20

Which makes focal distance c=15

For hyperbola

[tex]a^{2}+b^{2}=c^{2}[/tex]

2b is conjugate axis

2a is the transverse axis

So for this hyperbola 2b=20

                                     b=20/2

                                      b=10

and c=15

then

[tex]a=\sqrt{c^{2}-b^{2}  }\\ a=\sqrt{(15)^{2} -(10)^{2} }  \\a=\sqrt{25}\\ a=5[/tex]

that makes

2a=10 Distance between vertices

If P is point

So for this hyperbola

|PA-PB|=10

[tex](PA-PB)^{2} =100[/tex]

We know for ellipse

PA+PB=50

[tex](PA+PB)^{2}=2500[/tex]

Since

[tex](PA+PB)^{2}-(PA-PB)^{2}\\  4PA.PB=2500-100\\PA.PB=\frac{2400}{4}\\ PA.PB=600[/tex]

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