Find equations of both lines through the point (2, -3) that are tengetn to the parabola y=x^2+x
i got the derivative as y prime = 2x +1, and i don't know where to go from there. PLEASE HELP this is due tomorrow!!!
To find the equations of the tangent lines to the parabola y = x^2 + x that pass through the point (2, -3), you're on the right track by finding the derivative. The derivative gives you the slope of the tangent line at any given point on the parabola.
The derivative you found, y' = 2x + 1, represents the slope of the tangent line at any point (x, y) on the parabola.
Now, let's proceed to find the specific tangent lines that pass through the point (2, -3).
Step 1: Substitute the x and y values of the given point into the equation y = x^2 + x.
-3 = 2^2 + 2
-3 = 4 + 2
-3 = 6
This step shows that the given point (2, -3) does NOT lie on the parabola y = x^2 + x. Therefore, it must not be a point of tangency.
Step 2: Find the slope of the tangent lines using the derivative, y' = 2x + 1.
At x = 2, the slope of the tangent line is:
y' = 2(2) + 1
y' = 4 + 1
y' = 5
So, the slope of the tangent lines through the point (2, -3) is 5.
Step 3: Use the point-slope form of a linear equation to find the equations of the tangent lines.
The point-slope form of a linear equation is: y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope.
Equation of the first tangent line:
Using (2, -3) as the point and a slope of 5:
y - (-3) = 5(x - 2)
y + 3 = 5x - 10
y = 5x - 13
Equation of the second tangent line:
Using (2, -3) as the point and a slope of 5:
y - (-3) = 5(x - 2)
y + 3 = 5x - 10
y = 5x - 13
Therefore, the equations of both tangent lines through the point (2, -3) that are tangent to the parabola y = x^2 + x are:
y = 5x - 13
y = 5x - 13
Please note that the two equations are the same because both tangent lines have the same slope and pass through the same point, resulting in the same line.