Question:
Find the equation of the tangent line to the curve $y=x^{2}-2 x+7$ which is
perpendicular to the line $5 y-15 x=13$
Solution:
slope of given line is 3
finding the slope of the tangent by differentiating the curve
$\frac{d y}{d x}=2 x-2$
$m($ tangent $)=2 x-2$
since both lines are perpendicular to each other
$(2 \times-2) \times 3=-1$
$\mathrm{X}=\frac{5}{6}$
since this point lies on the curve, we can find y by substituting $x$
$y=\frac{25}{36}-\frac{10}{6}+7=\frac{217}{36}$
therefore, the equation of the tangent is
$y-\frac{217}{36}=-\frac{1}{3}\left(x-\frac{5}{6}\right)$
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