Find equation of two tangent lines to curve y=x^2 that intersect @ (2,-2). Need exact values.
To find the equations of two tangent lines to the curve y = x^2 that intersect at (2, -2), we need to follow these steps:
Step 1: Find the derivative of the curve y = x^2 using the power rule. The derivative represents the slope of the tangent line at any given point.
Let's differentiate y = x^2:
dy/dx = 2x
Step 2: Find the slope of the tangent lines at x = 2 by substituting x = 2 into the derivative.
dy/dx = 2x
When x = 2,
dy/dx = 2(2) = 4
So, the slope of the tangent lines at x = 2 is 4.
Step 3: Use point-slope form to find the equations of the tangent lines.
The point-slope form of a line is given by:
y - y1 = m(x - x1)
For the first tangent line:
Let's use the point (2, -2) and the slope m = 4 to get the equation:
y - (-2) = 4(x - 2)
y + 2 = 4(x - 2)
y + 2 = 4x - 8
y = 4x - 10
For the second tangent line, we use the same slope m = 4 but a different point:
Using (2, -2), the equation becomes:
y - (-2) = 4(x - 2)
y + 2 = 4(x - 2)
y + 2 = 4x - 8
y = 4x - 10
So, the equations of the two tangent lines to the curve y = x^2 that intersect at (2, -2) are:
y = 4x - 10