Find the equation of the tangent to the parabola $y = x^{2} - 7 x + 3$ , if that tangent is parallel to the straight line $5 x + y - 3 = 0.$

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Solution

The slope of tangent to curve $y = x^{2} - 7 x + 3$ parallel to $5 x + y - 3 = 0$ is $^{'} - 3^{'}$
$\frac{d y}{d x} = 2 x - 7 = s l o p e = - 5 \Rightarrow x = 1$
If $x_{1} = 1 t h e n y_{1} = 1^{2} - 7.1 + 3 = - 3$
The equation of line passing through $(1, - 3)$ and having slope $m = - 5$
$(x - 1) (- 5) = y + 3$
$5 x - y - 2 = 0$ is the tangent parallel to $5 x + y - 3 = 0$

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