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About LinkBacksShow 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![9x^2+4y^2\:=\:36\;\;{\color{blue}[2]}](http://latex.codecogs.com/png.latex?9x^2+4y^2\:=\:36\;\;{\color{blue}[2]})
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.
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