Co Ordinate Geometry

To find the point where the y-coordinate is changing thrice as fast as the x-coordinate, we need to analyze the relationship between the derivatives of y and x with respect to time.
Given the curve:

To find the derivative , differentiate the equation with respect to :

We are told that the y-coordinate changes thrice as fast as the x-coordinate, i.e.,

Using the chain rule, we know:

Equating:

Since , we can divide both sides by :

Substituting the value of :

Simplifying gives:

Substituting into the original equation to find :

Thus, the point on the curve where the y-coordinate is changing thrice as fast as the x-coordinate is:

To find the equation of the normal to the curve at , follow these steps:

Therefore, the equation of the normal to the curve at is .

To determine the coordinates of the foot of the perpendicular drawn from the origin to the plane , follow these steps:

Therefore, the coordinates of the foot of the perpendicular are .

To find the maximum slope of the curve , we must determine when the derivative reaches its maximum value. The derivative gives us the slope of the tangent at any point on the curve.

First, we find the derivative: . Using basic differentiation rules, we get:

.

Next, locate the critical points of the derivative by setting the second derivative to zero, as these points indicate maximum or minimum slopes:

.

This results in: .

Set the second derivative to zero to find critical points: . Solving gives: .

To determine if this point is a maximum, calculate the second derivative: , which indicates a maximum.

Substitute into the first derivative to find the maximum slope:

.

Therefore, the maximum slope of the curve is 11.