Let $E$ the ecllipse, with center $O = (0,0)$, focus $F=(4,0)$ and vertex $V=(5, 0)$. Let $N$ be a point on the ellipse $E$, and let $Q$ be the orthogonal projection of $N$ onto the y-axis. Find the coordinates of the intersection of lines $ON$ and $QV$.
Excuse me first, I'm writing from a cellphone. I find $E$, $Q$ is $(0,y)$ and $N=(x, y)$. Then i found the intersection between the two lines, but I didn't get anything. Any help?
HINTS
Y would say:
The parametric equation of E: $\displaystyle \quad x = 5\cos t,\quad y = 3 \sin t$
$\Rightarrow$ the koordinates of points $\displaystyle \quad N(5\cos t, 3\sin t), \quad Q(0,3\sin t)$
$\Rightarrow$ the equations of the lines $\displaystyle \quad ON: y = \frac{3}{5}\tan t\cdot x, \quad QV: \frac{x}{5}+\frac{y}{3\sin t}=1$
$\Rightarrow$ the coordinates of the intersection of lines ON and QV $\equiv$ the parametric equation of the locus:
$\displaystyle \quad x = \frac{5}{2}(1-\tan^2\frac{t}{2}), \quad y = 3\tan \frac{t}{2}$
$\Rightarrow$ the equation of the locus in Cartesian coordinates:
$\displaystyle \quad y^2=3.6(2.5-x)$