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**arze** it is given that the line y=mx+c is tangent to the ellipse

$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ if $\displaystyle a^2m^2=c^2-b^2$.

Show that if the line y=mx+c passes through the point $\displaystyle (\frac{5}{4}, 5)$ and is tangent to the ellipse $\displaystyle 8x^2+3y^2=35$, then c is $\displaystyle \frac{35}{3}$ or $\displaystyle \frac{35}{9}$.

At the point, $\displaystyle 5=\frac{5}{4}m+c$

substitute m,

$\displaystyle \frac{35}{3}(4-\frac{4}{5}c)^2=c^2-\frac{35}{8}$

i rearrange it and get

$\displaystyle 776c^2-8960c+22925=0$

but the roots of c are not what are given.

Thanks