The acute angle between the pair of tangents drawn to the ellipse 2x^2+3y^2=5 from the poi…

Question

The acute angle between the pair of tangents drawn to the ellipse $2x^2+3y^2=5$ from the point $(1,3)$ is

Accepted Answer

$S=2x^2+3y^2-5$; $S_1=2+27-5=24$; $T=2x+9y-5$. Pair of tangents: $24S-T^2=0$:
$44x^2-36xy-9y^2+20x+90y-145=0.$

Here $A=44,\ B=-9,\ H=-18$:
$$\tan\theta=\frac{2\sqrt{H^2-AB}}{A+B}=\frac{2\sqrt{324+396}}{35}=\frac{2\cdot12\sqrt5}{35}=\frac{24\sqrt5}{35}=\frac{24}{7\sqrt5}.$$

Solutions · 1