Show that the line with equation y=2x+5 is a tangent to the ellipse with equation 9x^2+4y^2=36.
Hello, rednest!
First, find their intersection(s).Show that the line with equation
is a tangent to the ellipse with equation
Substitute [1] into [2]: .
. . which simplifies: .
. . and factors: .
. . and has one root: .
Substitute into [1] and we get: .
The line and ellipse have one common point: .
That should be sufficient, but let's check their slopes.
Obviously, the line has slope 2.
Differentiate [2] implicitly: .
At , the tangent to the ellipse has slope: .
. . Their slopes are equal!
Therefore, the line is tangent to the ellipse.