Question

Find the equations of both of the tangent lines to the ellipse that pass through the point (x, y). List the line with the smaller slope first.

a) y=mx+b
b) y=−mx+b
c) y=nx+c
d) y=−nx+c

Answer 1

To find the tangent lines to an ellipse passing through an external point, the slope of the tangent lines at the points of tangency is needed, following which the equations can be expressed in the slope-intercept form y = mx + b.

Step-by-step explanation:

To find the equations of the tangent lines to the ellipse that pass through a given point (x, y), we need to follow a few steps. First, we must identify the knowns such as the coordinates of the point. Then we determine the slope of the tangent line using an applicable equation. In the case of an ellipse, this usually involves calculus, as we need to find the derivative of the ellipse equation at the tangent point.

Once we have determined the slope (m) of the tangent lines, we can use the slope-point form of a line, which is y - y1 = m(x - x1), where (x1, y1) is the point of tangency on the ellipse. However, in the absence of a specific ellipse equation and a given point (x, y), we cannot provide the exact equations of the tangent lines. But we can conclude that the linear equations will have the form of y = mx + b where m is the slope and b is the y-intercept.