I WILL GIVE BRAINLY
[tex]x^{2} +9y^{2} =1[/tex] is an ellipse. The line y=mx+2 is tangent to the ellipse. Determine the value of ([tex]m^{2}[/tex]). Express your answer as an integer or as a common or improper fraction.

The line will be tangent to the ellipse where there is one point of intersection between the line and the ellipse. We can find that case by looking at the discriminant of the quadratic.

x^2 +9(mx +2)^2 = 1

x^2 +9(m^2x^2 +4mx +4) = 1

(9m^2 +1)x^2+36mx +35 = 0 . . . . subtract 1, collect terms

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Then the discriminant is ...

(36m)^2 -4(9m^2 +1)(35)

We want it to be zero, so we have ...

1296m^2 -1260m^2 -140 = 0 . . . . eliminate parentheses

36m^2 = 140 . . . . . . . . . . . . . . . add 140

m^2 = 140/36 . . . . . . . . . . . . divide by 36

m^2 = 35/9

I WILL GIVE BRAINLY [tex]x^{2} +9y^{2} =1[/tex] is an ellipse. The line y=mx+2 is tangent to the ellipse. Determine the value of ([tex]m^{2}[/tex]). Express your answer as an integer or as a common or improper fraction.

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I WILL GIVE BRAINLY
[tex]x^{2} +9y^{2} =1[/tex] is an ellipse. The line y=mx+2 is tangent to the ellipse. Determine the value of ([tex]m^{2}[/tex]). Express your answer as an integer or as a common or improper fraction.