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P N is the ordinate of any point P on th...

`P N` is the ordinate of any point `P` on the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` and `AA '` is its transvers axis. If `Q` divides `A P` in the ratio `a^2: b^2,` then prove that `N Q` is perpendicular to `A^(prime)Pdot`

Text Solution

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Let coordinates of point P on hyperbola be `(a sec theta, b tan theta)`.
`therefore" N-=(a sec theta, 0)`
Since Q divides AP in the ratio `a^(2):b^(2)`, we have
`therefore" "Q-=((ab^(2)+a^(3)sectheta)/(a^(2)+b^(2)),(a^(2)b tantheta)/(a^(2)+b^(2)))`
`"Slope of A'P"=(b tan theta)/(a(sec theta+1))`
`"Slope of QN"=(a^(2)btantheta)/(ab^(2)+a^(3)sec theta-a^(3)sec theta-ab^(2)sectheta)`
`=(a^(2)btan theta)/(ab^(2)(1-sec theta))`
`"Slope of A'P"xx"Slope of QN"=(a^(2)b^(2)tan^(2)theta)/(-a^(2)b^(2)tan^(2)theta)=-1`
Hence, QN is perpendicular of A'P.
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