Final answer:
The equation of an ellipse is x²/a² + y²/b² = 1. When a tangent line to the ellipse intersects the x-axis and y-axis, it cuts off intercepts of length h and k, respectively.
The slope of the tangent line can be found by taking the derivative of the equation of the ellipse and evaluating it at the point of tangency.
Step-by-step explanation:
The equation for an ellipse is x²/a² + y²/b² = 1, where a represents the semi-major axis and b represents the semi-minor axis. When a tangent line is drawn to the ellipse and it intersects the x-axis and y-axis, it cuts off intercepts of length h and k, respectively.
Let's consider the point of tangency on the ellipse as (x₀, y₀). The slope of the tangent line can be determined by taking the derivative of the equation of the ellipse and evaluating it at (x₀, y₀). The slope of the tangent line is then given by m = -bx₀/ay₀. Using this slope, we can write the equation of the tangent line in point-slope form as y - y₀ = (-bx₀/ay₀)(x - x₀).
Now, we need to find the x-intercept and y-intercept of this line. To find the x-intercept, we set y = 0 in the equation of the tangent line and solve for x. Similarly, to find the y-intercept, we set x = 0 and solve for y.