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    Finding the tangent to a curve

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    Find the condition that the curve y = mx + c to be a tangent to the parabola y^2 = 4 ax and also determine the point of contact.

    © BrainMass Inc. brainmass.com December 24, 2021, 4:48 pm ad1c9bdddf
    https://brainmass.com/math/graphs-and-functions/finding-tangent-curve-8628

    SOLUTION This solution is FREE courtesy of BrainMass!

    Let (x1, y1) be the point where the line

    y = mx + c .........(1)

    touches the parabola y^2 = 4ax ....(2)

    But the equation of the tangent at (x1, y1) to the parabola is,

    yy1 = 2a(x+x1) .......(3)

    Since (1) is a tangent to the parabola, (1) and (3) represent the same line.

    therefore, on equating,

    y1 / 1 = 2a/m = 2 a x1 / c

    y1 = 2a/m and x1 = c/m .....(4)

    Since (x1, y1) is a point on the parabola(2),

    y1^2 = 4 a x1

    Substitute for x1 and y1 from the above relations,

    4a^2/m^2 = 4 a c/m

    or, c = a/m

    This is the required condition.

    Substituting this value of c, we get the co-ordinates of the point of contact as,

    (a/m^2, 2a/m)

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 4:48 pm ad1c9bdddf>
    https://brainmass.com/math/graphs-and-functions/finding-tangent-curve-8628

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