What are the equations of the tangent lines out P(2,1) to the parabola with equation y=x2? Thank you!

1 Answer
Ananda Dasgupta
Jun 11, 2018

Two tangents

  • y=2(2−√3)x−7+4√3
  • y=2(2+√3)x−7−4√3

Explanation:

The tangent to a curve y(x) at the point (x0,y0) is given by

y−y0=∣∣∣dydx∣∣∣x=x0(x−x0)

For y=x2 this becomes

y−y0=2x0(x−x0)

with y0=x20, so that

y=2x0x−x20

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

1=4x0−x−x20⇒
x20−4x0+1=0⇒
(x0−2)2=3⇒
x0=2±√3

So, there are two tangents to the parabola from (2,1) to the parabola y=x2 and they are

  • for x0=2−√3, y=2(2−√3)x−7+4√3
  • for x0=2+√3, y=2(2+√3)x−7−4√3
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