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Government$
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Homework Statement
Find parameter a so that line [itex]y=ax + 11[/itex] touches ellipse [itex]3x^2 + 2y^2 = 11[/itex]
The Attempt at a Solution
|I can rewrite ellipse equation like [itex]\frac{x^2}{\frac{11}{3}} + \frac{y^2}{\frac{11}{2}} = 1[/itex]
And i know that line [itex]y=kx + n[/itex] touches ellipse when [itex]a^2k^2 + b^2 = n^2[/itex]
So in essence i am looking for a slope of a line.
[itex]({\frac{11}{3}})^2k^2 + ({\frac{11}{2}})^2 = 11^2[/itex]
[itex]({\frac{121}{9}})k^2 + ({\frac{121}{4}}) = 121[/itex]
When i solve for k i get [itex]k^2 = 6.75[/itex]
Problem is that this is not a solution. Here is what my textbook says:
Line that touches ellpise if and only if system y=ax + 11, 3x^2 + 2y^2 = 11 has one solution i.e. when discriminant of quadratic equation 3x^2 + 2(ax+ 11)^2 = 11 is equal to 0, and for that [itex]a = \pm \sqrt{\frac{63}{2}}[/itex]
I tried graphing this problem with both solutions and line doesn't touches ellipse.