The length of the minor axis (along y-axis) of an ellipse in the standard form is 4/√3. If this ellipse touches the line, x + 6y = 8; then its eccentricity is :

Solution

Maths sir · 4 steps

Get b from the minor axis, then the tangency condition c² = a²m² + b² hands you a.

1. Find b. The minor axis lies along the y-axis, so the ellipse is x²/a² + y²/b² = 1 with a > b. The length of the minor axis is 2b = 4/√3, so b = 2/√3 and b² = 4/3.
2. Slope form. Rewrite the line as y = -1/6x + 4/3. So m = -1/6 and c = 4/3. A line y = mx + c touches the ellipse exactly when c² = a²m² + b².

   16/9 = a²/36 + 4/3 ⇒ a²/36 = 4/9 ⇒ a² = 16

3. Eccentricity. e² = 1 - b²/a² = 1 - (4/3)/16 = 1 - 1/12 = 11/12.
4. So. e = √(11/12) = 1/2√(11/3), which is option (d).

The idea to keep. Whenever a line touches an ellipse, put it in the form y = mx + c and use c² = a²m² + b². One known axis plus this condition gives the other axis.

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