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If A O Ba n dC O D are two straight line...

If `A O Ba n dC O D` are two straight lines which bisect one another at right angles, show that the locus of a points `P` which moves so that `P A*P B=P C*P D` is a hyperbola. Find its eccentricity.

Text Solution

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The correct Answer is:
`e=sqrt2`
Taking AOB and COD as the x- and y-axes and their point of intersection O as the origin, clearly, O is the midpoint of AB and CD. Let A be (a, 0) and C be (0, c). Then B is `(-a, 0)` and D is `(0, -c)`.

Let `P-=(x,y)`. Given that
`PA*PB=PC*PD`
`"or "sqrt((x-a)^(2)+y^(2))sqrt((x+a)^(2)+y^(2))=sqrt(x^(2)+(y-c)^(2))sqrt(x^(2)+(y+c)^(2))`
After simplification, we have
`x^(2)-y^(2)=((a^(2)-c^(2)))/(2)`
which is a rectangular hyperbola with eccentricity `sqrt2`.
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