Let a circle of radius 4 be concentric to the ellipse 15 x²+19 y²=285. Then the common tangents are inclined to the minor axis of the ellipse at the angle :

Solution

Maths sir · 4 steps

Write one tangent line that touches both curves, match the two conditions on c, and read off the slope.

1. Standard form. Divide by 285 to get x²/19 + y²/15 = 1. So a² = 19 and b² = 15. Since a² > b², the major axis is the x-axis and the minor axis is the y-axis.
2. Two tangency conditions. Take the common tangent as y = mx + c. It touches the ellipse when c² = 19m² + 15. It touches the circle x² + y² = 16 when c² = 16(1 + m²).

   19m² + 15 = 16m² + 16 ⇒ 3m² = 1 ⇒ m = ±1/√3

3. Angle with axes. A slope of ±1/√3 means the tangents make π/6 with the x-axis, which is the major axis. The y-axis is perpendicular to it, so the angle with the minor axis is π/2 - π/6 = π/3.
4. So. The common tangents are inclined to the minor axis at π/3, which is option (b).

The idea to keep. For a common tangent, write c² from each curve's tangency condition and set them equal. The slope gives the angle with the x-axis, so check which axis the question actually asks about.

Answer from the NTA final answer key. Solution by PrepWiser.