Find the slope of the tangent line to the ellipse x^2/36 + y^2/49 =1 at the point (x,y).

slope =_______

Are there any points where the slope is not defined? (Enter them as comma-separated ordered-pairs, e.g., (1,3), (-2,5). Enter none if there are no such points.)
slope is undefined at____

Please help..

Oh, ellipses can be quite slippery! But don't worry, I've got your back! To find the slope of the tangent line to the ellipse, we need to differentiate the equation with respect to x. Let's get into it!

Taking the derivative of the given equation, we get:
(2x/36) + (2y/49) * dy/dx = 0

Now, let's isolate dy/dx (the slope):
dy/dx = - (2x/36) * (49/2y)
dy/dx = - (7/6) * (x/y)

So, the slope of the tangent line at any point (x, y) on the ellipse is (-7x)/(6y).

Now, are there any points where the slope is not defined? Well, if the denominator of the slope expression becomes zero, the slope is undefined. In our case, if y = 0, the slope will be undefined. Therefore, the slope is undefined at the points where y = 0.

Therefore, the points where the slope is not defined are (0, 0).

I hope that clears things up, and remember, math can be funnier than you think!

To find the slope of the tangent line to the ellipse at the point (x, y), we can differentiate the equation of the ellipse implicitly.

Given the equation:
x^2/36 + y^2/49 = 1

Differentiating both sides with respect to x:
(2x/36) + (2y/49) * (dy/dx) = 0

Simplifying:
(2x/36) + (2y/49) * (dy/dx) = 0

Now, let's solve for dy/dx, which represents the slope of the tangent line at a given point (x, y):

(2y/49) * (dy/dx) = -(2x/36)

dy/dx = -(2x/36) / (2y/49)
dy/dx = -(49x) / (36y)

So, the slope of the tangent line to the ellipse at the point (x, y) is given by -(49x) / (36y).

Next, we will determine any points where the slope is undefined. The slope will be undefined where the denominator (36y) of the above expression becomes zero.

To find the points where the denominator is zero, we set:
36y = 0

Solving for y:
y = 0

Therefore, the slope is undefined at the point (x, y) where y = 0.

In summary:
The slope of the tangent line to the ellipse at the point (x, y) is -(49x) / (36y).
The slope is undefined at the point where y = 0.

To find the slope of the tangent line to the ellipse at a certain point (x, y), we can use calculus and the implicit differentiation approach. Here's a step-by-step explanation of how to find the slope:

Step 1: Start with the equation of the ellipse:
x^2/36 + y^2/49 = 1

Step 2: Differentiate both sides of the equation implicitly with respect to x:
(1/36)(2x) + (1/49)(2y)(dy/dx) = 0

Step 3: Simplify the equation:
(1/18)x + (2y/49)(dy/dx) = 0

Step 4: Solve for dy/dx to get the slope of the tangent line:
dy/dx = (-49/36)(x/y)

Now that we have the equation for the slope, we can substitute the values (x, y) to calculate the slope.

Regarding points where the slope is undefined, it happens at the points where the derivative dy/dx is undefined or does not exist. In this case, it occurs when y = 0 since division by zero is undefined. To find such points, we substitute y = 0 into the equation of the ellipse and solve for x:

x^2/36 + (0^2)/49 = 1
x^2/36 = 1
x^2 = 36
x = ±6

Therefore, there are two points, (6, 0) and (-6, 0), where the slope is undefined.

To summarize:
- The slope of the tangent line to the ellipse at the point (x, y) is dy/dx = (-49/36)(x/y).
- The slope is undefined at the points (6, 0) and (-6, 0).

Use implicit differentiation, by differentiating each side of the equation with respect to x.

x/18 + (2y/49)*dy/dx = 0
dy/dx = -(49/2)(x/18y)
= -(49/36)(x/y)

The slope is undefined where y = 0, which is where x = +/- 6:
(-6,0) and (6,0)