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If S ans S' are the foci, C is the cente...

If `S` ans `S'` are the foci, C is the center , and P is point on the rectangular hyperbola, show that `SP ×S'P=(CP)^2`

Text Solution

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Let any point P on the hyperbola `x^(2)-y^(2)=a^(2)be (x_(1),y_(1))`.
Now, SP `= ex_(1)-a and S'P=ex_(1)+a.` Then,
`SPxxS'P=e^(2)x_(1)^(2)-a^(2)`
`=2x_(1)^(2)-a^(2)`
`=2x_(1)^(2)-(x_(1)^(2)-y_(1)^(2))" "[because" Point" (x_(1),y_(1)) " lies on the hyperbola"]`
`x_(1)^(2)+y_(1)^(2)`
`CP^(2)`
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