Find the equation of the tangent line to the curve 2x^2y-3y^2=-11 at the point (2,-1).
2x^2y-3y^2=-11
4xy + 2x^2 dy/dx - 6y dy/dx = 0
dy/dx(2x^2 - 6y) = -4xy
dy/dx = -4xy/(2x^2 - 6y)
at (2,-1)
dy/dx = -4(2)(-1)/(8 -6(-1))
= 8/14 = 4/7
equation of tangent:
y+1 = (4/7)(x - 2)
massage it any way you need to
To find the equation of the tangent line to a curve at a given point, we need to follow these steps:
Step 1: Differentiate the equation implicitly with respect to x to find the derivative dy/dx.
Step 2: Substitute the x-coordinate of the given point into the derivative to find the slope of the tangent line.
Step 3: Use the point-slope form of a line to write the equation of the tangent line.
Let's go through these steps one by one:
Step 1: Differentiate the equation implicitly with respect to x to find the derivative dy/dx.
Differentiating the given equation implicitly with respect to x involves taking the derivative of each term with respect to x and applying the product rule and chain rule as needed. The equation 2x^2y - 3y^2 = -11 can be rewritten as 2x^2y - 3y^2 + 11 = 0. Differentiating both sides of this equation with respect to x gives:
(4xy + 2x^2 * dy/dx) - (6y * dy/dx) = 0
Rearranging the terms, we have:
2x^2 * dy/dx - 6y * dy/dx = -(4xy + 11)
Factoring out dy/dx:
dy/dx(2x^2 - 6y) = -(4xy + 11)
Simplifying further:
dy/dx = -(4xy + 11)/(2x^2 - 6y)
Step 2: Substitute the x-coordinate of the given point into the derivative to find the slope of the tangent line.
The given point is (2, -1), so we substitute x = 2 and y = -1 into the derivative:
dy/dx = -(4(2)(-1) + 11)/(2(2)^2 - 6(-1))
= -(8 + 11)/(8 + 6)
= -19/14
Therefore, the slope of the tangent line is -19/14.
Step 3: Use the point-slope form of a line to write the equation of the tangent line.
Using the point-slope form of a line (y - y1) = m(x - x1), where (x1, y1) is the given point and m is the slope we found, we substitute the values:
(y - (-1)) = (-19/14)(x - 2)
y + 1 = (-19/14)(x - 2)
This is the equation of the tangent line to the curve 2x^2y - 3y^2 = -11 at the point (2, -1).