Question:

Consider two straight lines, each of which is tangent to both the circle x² + y² = 1 and the parabola y² = 4x. Let these lines intersect at the point Q. Consider the ellipse whose center is at the origin O(0,0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is √2, then which of the following statement(s) is (are) TRUE?

The area of the region bounded by the ellipse between the lines x = 1/√2 and x = 1 is 1/4√2(π - 1)

The area of the region bounded by the ellipse between the lines x = 1/√2 and x = 1 is 1/16√2(π - 1)

For the ellipse, the eccentricity is 1/√2 and the length of the latus rectum is 1

For the ellipse, the eccentricity is 1/2 and the length of the latus rectum is 1/2

Solution:

Let equation of common tangent be y = mx + 1/m
∴ 0 + 0 + 1/m = √1 + m²
⇒ 1/m² = 1 + m² ⇒ m⁴ + m² - 1 = 0 ⇒ m = ±1
Equation of common tangents are y = x + 1 and y = -x + 1
Point Q is (0, 1) ⇒ Equation of ellipse is x²/1 + y²/1/2 = 1
(A) e = √1 - 1/2 = 1/√2 and LR = 2b²/a = 1
(C) Area = 2 ∫1/√21 √(1 - x²) dx = √2 [x/2 √(1 - x²) + 1/2 sin⁻¹x]1/√21 = √2 [π/4 - (1/4 + π/8)] = √2 (π/8 - 1/4) = π/4√2 - 1/2√2 = π/4√2 - 1/(2√2)
Correct answers are (A) and (C).