# Math Help - Finding tangent lines

1. ## Finding 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.

2. Originally Posted by yeloc
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.
Draw the parabola. Plot the point. Draw a line that is tangent to the parabola and that also passes through the plotted point. Your task is to find the equation of that line.

3. 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?

4. Originally Posted by yeloc
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.

No it isn't a trick question, it is asking for you to find all the lines that are tangents to y=x^2 that also pass through (2,3).

5. Originally Posted by yeloc
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.
the slope of any such line must be $m=2x$ and of the form ...

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

simplify ...

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

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

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

solve for x, then get the equations of the two lines that meet the conditions set forth in the problem.

6. Originally Posted by skeeter
the slope of any such line must be $m=2x$ and of the form ...

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

simplify ...

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

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

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

solve for x, then get the equations of the two lines that meet the conditions set forth in the problem.

Skeeter, I understand everything up to where you set $2x^2 -4x+3=x^2$

7. Originally Posted by yeloc
Skeeter, I understand everything up to where you set $2x^2 -4x+3=x^2$
when a tangent line touches a curve, the point of tangency is on the line and the curve at the same time ... i.e. line = curve

8. Which equation do you use to find the y values after getting the two x values?

9. Originally Posted by yeloc
Which equation do you use to find the y values after getting the two x values?
y = x^2.