Find the condition for the following set of curves to intersect orthogonally: `(x^2)/(a^2)-(y^2)/(b^2)=1` and `x y=c^2` `(x^2)/(a^2)+(y^2)/(b^2)=1` and `(x^2)/(A^2)-(y^2)/(B^2)=1.`

Explanation

(i)`frac{x^{2}}{a^{2}}-frac{y^{2}}{b^{2}}=1 ldots(1) `

`x y=c^{2} ldots(2)`

Let the curves intersect orthogonally at `(x_{1}, y_{1})`.

On differentiating ( 1 ) on both sides w.r.t.x, we get


`frac{2 x}{a^{2}}-frac{2 y}{b^{2}} frac{d y}{d x}=0 `

`Rightarrow frac{d y}{d x}=frac{x b^{2}}{a^{2} y} `

`Rightarrow m_{1}=(frac{d y}{d x})_{(x_{1}, y_{1})}=frac{x_{1} b^{2}}{a^{2} y_{1}}`

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