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If P(x(1),y(1)) is a point on the hyperb...

If `P(x_(1),y_(1))` is a point on the hyperbola `x^(2)-y^(2)=a^(2)`, then `SO.S'P=` . . . .

A

`(x_(1)^(2)-y_(1)^(2))/(a^(2))`

B

`(x_(1)^(2)+y_(1)^(2))/(a^(2))`

C

`x_(1)^(2)-y_(1)^(2)`

D

`x_(1)^(2)+y_(1)^(2)`

Text Solution

Verified by Experts

The correct Answer is:
D
Given, equation of hyperbola
`x^(2)-y^(2)=a^(2)` ...(i)
If `P(x_(1),y_(1))` is a point on Eq. (i), then
`x_(1)^(2)-y_(1)^(2)=a^(2)` ....(ii)
Now, `SP=ex_(1)-a`
and `SP'=ex_(1)+a`
`thereforeSPdotSP'=(ex_(1)-a)(ex_(1)+a)`
`=e^(2)dotx_(1)^(2)-a^(2)`
`=2x_(1)^(2)-a^(2)` (since, for `x^(2)-y^(2)=a^(2),e=sqrt2`)
`=2x_(1)^(2)-(x_(1)^(2)-y_(1)^(2))` [using Eq. (i)]
`=x_(1)^(2)+y_(1)^(2)`
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Knowledge Check

  • If P(x_(1),y_(1)) is a point on the hyperbola x^(2)-y^(2)=a^(2) , then SP.S'P= . . . .

    A
    `(x_(1)^(2)-y_(1)^(2))/(a^(2))`
    B
    `(x_(1)^(2)-y_(1)^(2))/(a^(2))`
    C
    `x_(1)^(2)-y_(1)^(2)`
    D
    `x_(1)^(2)-y_(1)^(2)`

  • If P(theta) is a point on the hyperbola (x^(2))/(a^(2)) - (y^(2))/(b^(2)) = 1 , whose foci are S and S', then SP.S'P =

    A
    `a^(2) sec^(2) theta + b^(2) tan^(2) theta `
    B
    `a^(2) sec^(2) theta - b^(2) tan^(2) theta `
    C
    `a^(2) tan^(2) theta - b^(2) sec^(2) theta `
    D
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    B
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    C
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    D
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