How do you find the equation of the chord of contact to the parabola x^2=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(x-x_0) + y_0

where (x_0,y_0) = (3,-2)

y = m(x-3) -2" [1]"

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

y = x^2/8" [2]"

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

dy/dx = x/4

m = x/4" [3]"

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

x^2/8 = x/4(x-3) -2

Solve the above equation for the values of x:

x^2 = 2x(x-3) -16

x^2 = 2x^2-6x -16

0 = x^2-6x-16

0 = (x+2)(x-8)

x_1 = -2 and x_2 = 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:

y_1 = (-2)^2/8 and y_2 = 8^2/8

y_1 = 1/2 and y_2 = 8

The equation of the chord of contact is the equation of the line that connects the points (-2,1/2) and (8,8).

Compute the slope:

m = (8-1/2)/(8--2)

m = 3/4

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

y = 3/4(x-8)+8

y = 3/4x+2, -2 <= x<= 8

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:

![www.desmos.com/calculator](useruploads.socratic.orguseruploads.socratic.org)