How do you find the equation of the chord of contact to the parabola x2=8y from the point (3,-2)?

1 Answer
Jun 24, 2018

Start with the point-slope form of the equation of a line:

y=m(xx0)+y0

where (x0,y0)=(3,2)

y=m(x3)2 [1]

Express the equation of the parabola as y in terms of x:

y=x28 [2]

We know that the equation for m, at the tangents, the first derivative with respect to x:

dydx=x4

m=x4 [3]

Substitute equations [2] and [3] into equation [1]:

x28=x4(x3)2

Solve the above equation for the values of x:

x2=2x(x3)16

x2=2x26x16

0=x26x16

0=(x+2)(x8)

x1=2 and x2=8

The above are the x-coordinates of the two points of tangency originating from point (3,2).

Use equation [2] to find the corresponding y values:

y1=(2)28 and y2=828

y1=12 and y2=8

The equation of the chord of contact is the equation of the line that connects the points (2,12) and (8,8).

Compute the slope:

m=81282

m=34

Use the point-slope form of the equation of a line and the point (8,8)

y=34(x8)+8

y=34x+2,2x8

The above is the slope-intercept form of the equation of the chord of contact.

The following is a drawing of the parabola, the tangent lines, and the chord of contact:

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