# Finding tangent lines

• Sep 20th 2009, 03:04 PM
yeloc
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.
• Sep 20th 2009, 03:06 PM
mr fantastic
Quote:

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.
• Sep 20th 2009, 03:09 PM
yeloc
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 PM
hairymclairy
Quote:

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).
• Sep 20th 2009, 03:14 PM
skeeter
Quote:

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 \$\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 PM
yeloc
Quote:

Originally Posted by skeeter
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.

Skeeter, I understand everything up to where you set \$\displaystyle 2x^2 -4x+3=x^2\$
• Sep 20th 2009, 03:23 PM
skeeter
Quote:

Originally Posted by yeloc
Skeeter, I understand everything up to where you set \$\displaystyle 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
• Sep 20th 2009, 03:44 PM
yeloc
Which equation do you use to find the y values after getting the two x values?
• Sep 20th 2009, 03:56 PM
mr fantastic
Quote:

Originally Posted by yeloc
Which equation do you use to find the y values after getting the two x values?

y = x^2.