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A normal to the hyperbola (x^(2))/(a^(2)...

A normal to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` meets the axes in M and N and lines MP and NP are drawn perpendicular to the axes meeting at P. Prove that the locus of P is the hyperbola `a^(2)x^(2)-b^(2)y^(2)=(a^(2)+b^(2))^(2)`

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Knowledge Check

  • A tangent to the hyperbola (x^(2))/(4)-(y^(2))/(2)=1 1 meets x axis at p and y axis at Q line PR and QR are drawn such that OPRQ is rectangle where O is the origin then R lies on

    A
    `(4)/(x^(2))+(2)/(y^(2))=1`
    B
    `(2)/(x^(2))-(4)/(y^(2))=1`
    C
    `(2)/(x^(2))+(4)/(y^(2))=-1`
    D
    `(4)/(x^(2))-(2)/(y^(2))=1`

  • The normal at P(x_(1),y_(1)) on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 meets the coordinate axes at A and B. If O, is u b the origin and e, the eccentricity of the hyperbola, then

    A
    `OA=e^(2)x_(1)`
    B
    `OB=e^(2)x_(1)`
    C
    `OA=e^(2)y_(1)`
    D
    `OB=e^(2)x_(1)`

  • The tangent at P on the hyperbola (x^(2))/(a^(2)) -(y^(2))/(b^(2))=1 meets one of the asymptote in Q. Then the locus of the mid-point of PQ is

    A
    `3((x^(2))/(a^(2))-(y^(2))/(b^(2)))=4`
    B
    `(x^(2))/(a^(2))-(y^(2))/(b^(2)) =2`
    C
    `(x^(2))/(a^(2)) -(y^(2))/(b^(2)) =(1)/(2)`
    D
    `4((x^(2))/(a^(2))-(y^(2))/(b^(2)))=3`