At what rate is the particle’s distance to the point (1,1) changing at any time?

Question

A particle is moving around the ellipse 4x^2 + 16x^2 = 64 . At any time t, its x- and y- coordinates are given by x(t) = 4 cos t , and y(t) = 2 sin t . At what rate is the particle’s distance to the point (1,1) changing at any time? At what rate is the distance changing when t = pi/ 6 ?

Answer 1

Step 1:

-coordinate is .

-coordinate is .

The distance from the point is .

.

.

Step 2:

The rate of change in distance with respect to is .

Apply derivative on each side with respect to .

The rate of change at .

Find and .

.

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Contd...

Step 3:

The rate of change in distance at is

Substitute and in above expression.

.

The rate of change in distance at is .

Solution:

The rate of change in distance at is .