Show that the line with equation y=2x+5 is a tangent to the ellipse with equation 9x^2+4y^2=36.

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- Mar 13th 2008, 08:44 PMrednestCoordinate systems help!
Show that the line with equation y=2x+5 is a tangent to the ellipse with equation 9x^2+4y^2=36.

- Mar 13th 2008, 09:56 PMSoroban
Hello, rednest!

Quote:

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.