Find the equation of the tangent to the curve x^2+y^2=25 at (-3,4).
I'm really confused on how to get the answer, someone please help
Take a look at the Related Questions below. The general outline for solution is:
find the slope of the tangent to the curve at any x: dy/dx
evaluate y' at the given x. That will provide the slope of the line you want.
Now you have a point and a slope. Use the point-slope form of the equation for a line.
easy peasy . . .
unknown
To find the equation of the tangent to a curve at a given point, you need to determine the slope of the tangent line at that point. Then, you can substitute the point and the slope into the equation of a line to find the equation of the tangent.
Here's how you can solve the problem step-by-step:
1. Start with the equation of the curve: x^2 + y^2 = 25.
2. Take the derivative of both sides of the equation with respect to x to find the slope of the tangent line at any given point on the curve. Remember that the derivative of x^2 is 2x, and the derivative of y^2 with respect to x (dy/dx) is 2yy'.
Differentiating x^2 + y^2 = 25 with respect to x, we get:
2x + 2yy' = 0.
3. Solve this equation for the derivative of y (y') to find the slope of the tangent at any point (x, y) on the curve:
2yy' = -2x,
y' = -x/y.
4. Substitute the given point (-3, 4) into the derived slope equation, to find the slope of the tangent at that point:
y' = -(-3)/4 = 3/4.
5. Now that you have the slope of the tangent (m) and the given point (x1, y1) = (-3, 4), you can use the point-slope form of the equation of a line to find the equation of the tangent:
y - y1 = m(x - x1).
Substituting the values, we get:
y - 4 = (3/4)(x + 3).
6. Simplify the equation by multiplying through by 4 to eliminate the fraction:
4y - 16 = 3x + 9.
7. Rearrange the equation to get it in the standard form:
3x - 4y = -25.
So, the equation of the tangent to the curve x^2 + y^2 = 25 at the point (-3, 4) is 3x - 4y = -25.