Question

Find the equation of the tangent lines to the ellipse (x-3)²/25+(y-5)²/9=1 that have a slope of -1 .

Answer 1

To find the equations of the tangent lines with a slope of -1 to a given ellipse, we compute the partial derivatives to find the points where the slope of the tangent equals -1 and then write the equations for those tangent lines using the points found.

Step-by-step explanation:

To find the equations of the tangent lines to the ellipse (x-3)²/25+(y-5)²/9=1 with a slope of -1, we need to use the derivative of the ellipse to determine the points at which the slope of the tangent is -1.

1. First, we write the equation of the ellipse in implicit form: f(x, y) = (x-3)²/25 + (y-5)²/9 - 1 = 0.
2. Next, we find the gradient of the ellipse by taking the partial derivatives of f(x, y) with respect to x and y.
3. Using the condition that the slope of the tangent line is -1, we can set up the equation ²f/²y / ²f/²x = -1.
4. After solving for x and y, we determine the corresponding points on the ellipse.
5. Last, we use these points to write the equations for the tangent lines, making sure they have a slope of -1.