Question

Let (a, ẞ) be a point from which two perpendicular tangents can be drawn to the ellipse 4x² +5y² = 20. If F = 4α +3ẞ, then______.

Options

• \[-15\leq F\leq15\]

• \[F\geq0\]

• \[-5\leq F\leq20\]

• \[F\leq-5\sqrt{5}\mathrm{~or~}F\geq5\sqrt{5}\]

Answer 1

Let (a, ẞ) be a point from which two perpendicular tangents can be drawn to the ellipse 4x² +5y² = 20. If F = 4α +3ẞ, then \[-15\leq F\leq15\].

Explanation:

\[\left(a\right)\left(\alpha,\beta\right)\] lies on the director circle of the ellipse i.e. on \[x^{2}+y^{2}=9.\].

So, we can assume

\[\alpha=3\cos\theta,\beta=3\sin\theta\]

\[\therefore F=12\cos\theta+9\sin\theta-3(4\cos\theta+3\sin\theta)\]

\[\Rightarrow\quad-15\leq F\leq15\]