Question
The range of values of 'a' for which the common tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ and the parabola $y^2=4x$ and their chord of contact can form an equilateral triangle is_______.
I know that it is not appreciated to ask here without uploading any work, but, I am not able to think any thing on this. I am not able to approach this question. Any hints and suggestions are welcome.
Thanks for your effort and time.
Let $(x_1,y_1), (x_2,y_2)$ be the points of contact of the common tangent on the ellipse and parabola respectively, above the $x$ axis. Note that since the curves are symmetric with respect to the $x$ axis, the other two points of contact below the $x$ axis are given by $(x_1,-y_1),(x_2,-y_2)$. Again, because of symmetry, the $x$ axis is the angle bisector of the common tangents. Since we require the angle between the tangents to be $60\deg$, the tangents are inclined at $30\deg$ above and below the $x$ axis, their slopes given by $\tan(\pm30\deg)=\pm1/\sqrt3$.
Differentiate the equation of the parabola with respect to $x$ and set $y'=1/\sqrt3$,
$\displaystyle y^2=4x\implies yy'=2\implies y_2=2\sqrt3, x_2=3$
The equation of the common tangent is $\displaystyle\frac{y-2\sqrt3}{x-3}=\frac1{\sqrt3}$ and its intercepts on the axes are $(-3,\sqrt3)$.
The common tangent should intersect the ellipse only once.
$\implies |b|< \sqrt3, |a|<3$