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Find the point on the hyperbola x^2-9y^2...

Find the point on the hyperbola `x^2-9y^2=9` where the line `5x+12 y=9` touches it.

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Solving `5x+12y=9 or y=(9-5x)/(12) and x^(2)-9y^(2)=9,` we have
`x^(2)-9((9-5x)/(12))=9`
`"or "x^(2)-(1)/(16)(9-5x)^(2)=9`
`"or "16x^(2)-(25x^(2)-90x+81)=144`
`"or "9x^(2)-(25x^(2)-90x+81)=144`
`"or "9x^(2)-90x+225=0`
`"or "x^(2)-10x+25=0`
`"or "x=5`
`rArr" "y=(9-25)/(12)=-(4)/(3)`
So, point of contact is `(5,-(4)/(3))`.
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