# Thread: Tangent of ellipse problem

1. ## Tangent of ellipse problem

it is given that the line y=mx+c is tangent to the ellipse
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ if $a^2m^2=c^2-b^2$.
Show that if the line y=mx+c passes through the point $(\frac{5}{4}, 5)$ and is tangent to the ellipse $8x^2+3y^2=35$, then c is $\frac{35}{3}$ or $\frac{35}{9}$.

At the point, $5=\frac{5}{4}m+c$
substitute m,
$\frac{35}{3}(4-\frac{4}{5}c)^2=c^2-\frac{35}{8}$
i rearrange it and get
$776c^2-8960c+22925=0$
but the roots of c are not what are given.
Thanks

2. $a^2 = \frac{35}{8} ; and ; b^2 = \frac{35}{3}$

3. Originally Posted by sa-ri-ga-ma
$a^2 = \frac{35}{8} ; and ; b^2 = \frac{35}{3}$
yes. isn't that what i am using?

4. Originally Posted by arze
yes. isn't that what i am using?
No. You have interchanged a and b.

5. Originally Posted by arze
it is given that the line y=mx+c is tangent to the ellipse
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ if $a^2m^2=c^2-b^2$.
Show that if the line y=mx+c passes through the point $(\frac{5}{4}, 5)$ and is tangent to the ellipse $8x^2+3y^2=35$, then c is $\frac{35}{3}$ or $\frac{35}{9}$.

At the point, $5=\frac{5}{4}m+c$
substitute m,
$\frac{35}{3}(4-\frac{4}{5}c)^2=c^2-\frac{35}{8}$
i rearrange it and get
$776c^2-8960c+22925=0$
but the roots of c are not what are given.
Thanks
hi

like sarigama suggested , you misplaced both a^2 and b^2

should be

$
\frac{35}{8}(\frac{20-4c}{5})^2=c^2-\frac{35}{3}
$

6. ok, i see what you mean, but shouldn't a be bigger than b in the case of an ellipse?

7. Originally Posted by arze
ok, i see what you mean, but shouldn't a be bigger than b in the case of an ellipse?
Not necessarily.
If the foci lie on the y axis, a<b.