Final answer:
To find the equation of the normal to the ellipse 8x^2 + 5y^2 = 13 at (2,-7), differentiate the equation with respect to x to find the slope of the tangent. The slope of the normal is the negative reciprocal of the slope of the tangent. Use the equation y - y₁ = m(x - x₁) to find the equation of the normal.
Step-by-step explanation:
To find the equation of the normal to the ellipse 8x2 + 5y2 = 13 at the point (2,-7), we first need to find the slope of the tangent to the ellipse at that point. The tangent to an ellipse at a given point is perpendicular to the normal at that point. So, the slope of the normal will be the negative reciprocal of the slope of the tangent.
To find the slope of the tangent, we differentiate the equation of the ellipse with respect to x to get 16x + 10yy' = 0. Then, we substitute the x and y values of the given point (2,-7) into this equation and solve for y'.
After finding the slope of the tangent, we can determine the slope of the normal by taking its negative reciprocal. Finally, we can use the equation y - y₁ = m(x - x₁), where (x₁, y₁) is the point of tangency, to find the equation of the normal.