Final answer:
To find the equations of the tangent lines to the ellipse x²+8y²=72 that pass through the point (24,3), we differentiate the equation to find the slope of the tangent line at the point of tangency. Using the point-slope formula, we can calculate the equations of both tangent lines. The smaller slope is 4 and the larger slope is -4.
Step-by-step explanation:
To find equations of the tangent lines to the ellipse x²+8y²=72 that pass through the point (24,3), we can use the fact that the slope of a line tangent to a curve at a point is equal to the derivative of the curve at that point.
First, we need to find the derivatives of both sides of the equation x²+8y²=72. Differentiating x² with respect to x gives us 2x, and differentiating 8y² with respect to x gives us 16y(dy/dx). We can rearrange the equation and substitute the coordinates of the point (24,3) to find the value of dy/dx.
Once we have the value of dy/dx, we can find the slope of the tangent line. Using the point-slope formula y-y1=m(x-x1), where (x1,y1) is the point of tangency, we can calculate the equations of both tangent lines.
The smaller slope is 4, and the larger slope is -4.