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

1. Simplify the Ellipse Equation

The given equation is $$15x^2 + 19y^2 = 285$$. Divide the entire equation by $$285$$ to bring it into standard form:

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

• Semi-major axis squared ($$a^2$$): $$19$$
• Semi-minor axis squared ($$b^2$$): $$15$$
• The minor axis of this ellipse lies along the y-axis.

2. Identify the Circle Equation

The circle is concentric with the ellipse (centered at $$(0,0)$$) and has a radius of $$4$$:

$$x^2 + y^2 = 16 \implies r^2 = 16$$

3. Set the Condition for Common Tangents

A line $$y = mx + c$$ is tangent to:

• The ellipse if $$c^2 = a^2m^2 + b^2$$
• The circle if $$c^2 = r^2(1 + m^2)$$

Equating the two expressions for $$c^2$$:

$$19m^2 + 15 = 16(1 + m^2)$$

$$19m^2 + 15 = 16 + 16m^2$$

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

4. Determine the Angle of Inclination

The slope $$m$$ represents the tangent of the angle $$\theta$$ that the line makes with the x-axis:

$$\tan \theta = \frac{1}{\sqrt{3}} \implies \theta = \frac{\pi}{6}$$

The question specifically asks for the angle of inclination to the minor axis (the y-axis):

$$\text{Angle with y-axis} = \frac{\pi}{2} - \theta = \frac{\pi}{2} - \frac{\pi}{6} = \frac{\pi}{3}$$

Correct Option: (A) $$\frac{\pi}{3}$$