Question

Find the slope of the line tangent to the following polar curve at the given point. At the point where the curve intersects the origin (if this occurs), find the equationn of the tangent line in polar coordinates. r = 9 + 7 cos theta; (16,0) and (2, pi) Find the slope of the line tangent to r = 9 + 7 cos theta at (16,0). Select the correct choice below and fill in any answer boxes within your choice. Find the slope of the line tangent to r = 9 + 7 cos theta at (2, pi). Select the correct choice below and fill in any answer boxes within your choice. At the point where the curve intersects the origin (if this occurs), find the equationn of the tangent line in polar coordinates. Select the correct choice below and fill in any answer boxes within your choice. The equationn of the tangent line when the curve intersects the origin is The curve does not intersect the origin.

Answer 1

To find the slope of the line tangent to the polar curve r = 9 + 7 cos(theta) at the point (16, 0), we can use the formula:

dy/dx = (dy/dtheta) / (dx/dtheta) = (r' sin(theta) + r cos(theta)) / (r' cos(theta) - r sin(theta))

where r' = dr/dtheta.

First, we need to find r' by taking the derivative of r with respect to theta:

r' = dr/dtheta = -7 sin(theta)

Then, we can plug in the given values to find the slope at (16, 0):

dy/dx = [(r' sin(theta) + r cos(theta)] / [r' cos(theta) - r sin(theta)]

= [(-7 sin(0) sin(0) + (9 + 7 cos(0)) cos(0))] / [(-7 sin(0) cos(0)) - (9 + 7 cos(0)) sin(0))]

= (9 + 7) / (-9) = -2

Therefore, the slope of the line tangent to the polar curve r = 9 + 7 cos(theta) at the point (16, 0) is -2.

To find the slope of the line tangent to the polar curve r = 9 + 7 cos(theta) at the point (2, pi), we can use the same formula as above:

dy/dx = (r' sin(theta) + r cos(theta)) / (r' cos(theta) - r sin(theta))

First, we need to find r' by taking the derivative of r with respect to theta:

r' = dr/dtheta = -7 sin(theta)

Then, we can plug in the given values to find the slope at (2, pi):

dy/dx = [(r' sin(theta) + r cos(theta)] / [r' cos(theta) - r sin(theta)]

= [(-7 sin(pi) sin(2) + (9 + 7 cos(pi)) cos(2))] / [(-7 sin(pi) cos(2)) - (9 + 7 cos(pi)) sin(2))]

= (-2) / (7)

Therefore, the slope of the line tangent to the polar curve r = 9 + 7 cos(theta) at the point (2, pi) is -2/7.

The polar curve r = 9 + 7 cos(theta) intersects the origin when r = 0, which occurs when cos(theta) = -9/7, which is not possible since the range of cosine function is [-1, 1]. Therefore, the curve does not intersect the origin.

Since the curve does not intersect the origin, the answer is "The curve does not intersect the origin" for the equation of the tangent line in polar coordinates.