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Hyperbola — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsHyperbola4 Marks
Question
If P is any point on the hyperbola whose axis are equal, prove that SP . SP =$\text{CP}^{2}$
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Answer

For a hyperbola if the lenght of semi transverse and semi conjugate axes are equal.
Then $\alpha\text{=b}$
Equation of the given hyperbola is
$\text{x}^{2}-\text{y}^{2}=\alpha^{2}.....(1)$
Then e $=\sqrt{2}, \text{C}=(0, 0), S=(\sqrt{2\text{a}}, 0), S=(-\sqrt(\text{2a}, 0)$
Let coordinates of any point P on hyperbola be $ (\alpha, \beta). $ Since P lies on (1)
? $\alpha-\beta^{2}=\alpha^{2}......(2)$
Now $\text{SP}^{2}.\text{SP}^{2}$ $=(2\alpha^{2}+\alpha^{2}+\beta^{2})^{2}-8\text{a}^{2}\alpha^{2}$
$=4\alpha^{4}+4\alpha^{2}(\alpha^{2}+\beta^{2})+(\alpha^{2}+\beta^{2})-8\text{a}^{2}\text{a}^{2}$
$=4\text{a}^{2}(\alpha^{2}-2\alpha^{2})+4\text{a}^{2}(\alpha^{2}+\beta^{2})+(\alpha^{2}+\beta^{2})^{2}$
$=4\alpha^{2}(\alpha^{2}-\beta^{2}-2\alpha^{2})+4\text{a}^{2}(\alpha^{2}+\beta^{2})+(\alpha^{2+}\beta^{2})^{2}$
$=(\alpha^{2}+\beta^{2})^{2}=\text{CP}^{4}$
$\text{SP. }\text{SP}=\text{CP}^{2}$

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