Find the points on the curve `x^2 + y^2 = 13`, the tangent at each one of which is parallel to the line `2x + 3y = 7`.

Answer

Points are `(2,3)` and `(-2,-3)`

Explanation

Let `(x_{1}, y_{1})` be the point.

The slope of line `2 x+3 y=7` is `\frac{-2}{3}`.


As point lies on the curve.


`x_{1}^{2}+y_{1}^{2}=13 \ldots` (1)


`x^{2}+y^{2}=13`


On differentiating both sides w.r.t `x`


`2 x+2 y \frac{d y}{d x}=0` `\Rightarrow \frac{d y}{d x}=\frac{-x}{y}`


Slope of the tangent at `(x_{1}, y_{1})=(\frac{d y}{d x})_{(x_{1}, y_{1})}=\frac{-x_{1}}{y_{1}}`


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