# 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.

Â© BrainMass Inc. brainmass.com December 24, 2021, 4:48 pm ad1c9bdddfhttps://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)

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