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Math Help - Finding tangent lines

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    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.
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    Quote Originally Posted by yeloc View Post
    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.
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    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?
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    Quote Originally Posted by yeloc View Post
    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).
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    Quote Originally Posted by yeloc View Post
    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.
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    Quote Originally Posted by skeeter View Post
    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
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    Quote Originally Posted by yeloc View Post
    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
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    Which equation do you use to find the y values after getting the two x values?
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    Quote Originally Posted by yeloc View Post
    Which equation do you use to find the y values after getting the two x values?
    y = x^2.
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