What are the equations of the tangent lines out P(2,1) to the parabola with equation $y = x^2$ ? Thank you!

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Answer 1 · 2018-06-11T01:11:40

Two tangents

The tangent to a curve $y(x)$ at the point $(x_0,y_0)$ is given by

$y-y_0 = |dy/dx|_{x=x_0}(x-x_0)$

For $y=x^2$ this becomes

$y-y_0 = 2x_0 (x-x_0)$

with $y_0 = x_0^2$ , so that

$y = 2x_0 x-x_0^2$

For the tangent to go through $(2,1)$ we must have

$1 = 4x_0-x-x_0^2 implies$
$x_0^2-4x_0+1 = 0 implies$
$(x_0-2)^2=3 implies$
$x_0 = 2 pm sqrt 3$

So, there are two tangents to the parabola from $(2,1)$ to the parabola $y=x^2$ and they are

What are the equations of the tangent lines out P(2,1) to the parabola with equation y = x^2y = x^2? Thank you!