Let a circle of radius 4 be concentric to the ellipse 15 x 2 + 19 y 2 = 285 . Then the common tangents are inclined to the minor axis of the ellipse at the angle [2023]

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Solution

Q.
Let a circle of radius 4 be concentric to the ellipse $15 x^{2} + 19 y^{2} = 285 .$ Then the common tangents are inclined to the minor axis of the ellipse at the angle [2023]

1 $\frac{π}{12}$

2 $\frac{π}{4}$

3 $\frac{π}{6}$

4 $\frac{π}{3}$

Ans.
(4)

$Given ellipse is \frac{x^{2}}{19} + \frac{y^{2}}{15} = 1$

$Equation of tangent to the ellipse is given by$

$y = m x \pm \sqrt{19 m^{2} + 15}$ ...(i)

$Equation of tangent to the circle x^{2} + y^{2} = 16 is given by$

$y = m x \pm 4 \sqrt{1 + m^{2}}$ ...(ii)

$From (i) and (ii), m x \pm 4 \sqrt{1 + m^{2}} = m x \pm \sqrt{19 m^{2} + 15}$

$\Rightarrow 4 \sqrt{1 + m^{2}} = \sqrt{19 m^{2} + 15}$ $\Rightarrow 16 (1 + m^{2}) = 19 m^{2} + 15$

$\Rightarrow 16 + 16 m^{2} = 19 m^{2} + 15$

$\Rightarrow 1 = 3 m^{2} \Rightarrow m^{2} = \frac{1}{3} \Rightarrow m = \pm \frac{1}{\sqrt{3}}$

$\Rightarrow \tan θ = \pm \frac{1}{\sqrt{3}} \Rightarrow θ = \frac{π}{6} with x-axis$

$∴$ $Common tangent inclined to the minor axis of the ellipse at the angle \frac{π}{3} .$

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