Final answer:
To find the equation of the tangent line to a parabola at specific points, solve for the points of intersection, calculate the slope at those points, and apply the point-slope form of a line using the point and slope.
Step-by-step explanation:
Finding the Equation of the Tangent to a Parabola
To find the equation of the tangent line to the curve y = x² - 2x - 3 at the points where the line y = 5 cuts the parabola, follow these steps:
Find the points of intersection between the parabola and the line y = 5 by setting the equations equal to each other and solving for x.
Calculate the slope of the parabola at the points of intersection using the derivative y' = 2x - 2.
Use the point-slope form of the line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point of intersection, to find the equation of the tangent line.
The slope of a curve at a point is the same as the slope of the tangent line at that point.