Find an equation for all lines which pass through the point (2,3) and are tangent to the parabola y=x^2

This is a trick question because the point (2,3) is not on the parabola.

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- Sep 20th 2009, 03:04 PMyelocFinding tangent lines
Find an equation for all lines which pass through the point (2,3) and are tangent to the parabola y=x^2

This is a trick question because the point (2,3) is not on the parabola. - Sep 20th 2009, 03:06 PMmr fantastic
- Sep 20th 2009, 03:09 PMyeloc
So there should be two tangent lines right? One with a positive slope and the other with a negative slope?

How do you determine where the tangent line crosses the parabola? - Sep 20th 2009, 03:11 PMhairymclairy
- Sep 20th 2009, 03:14 PMskeeter
the slope of any such line must be $\displaystyle m=2x$ and of the form ...

$\displaystyle y - 3 = 2x(x - 2)$

simplify ...

$\displaystyle y = 2x^2 - 4x + 3$

since each tangent line also touches the the parabola ...

$\displaystyle 2x^2 - 4x + 3 = x^2$

solve for x, then get the equations of the**two**lines that meet the conditions set forth in the problem. - Sep 20th 2009, 03:18 PMyeloc
- Sep 20th 2009, 03:23 PMskeeter
- Sep 20th 2009, 03:44 PMyeloc
Which equation do you use to find the y values after getting the two x values?

- Sep 20th 2009, 03:56 PMmr fantastic