Question

The number of following statements which are true in the case of the ellipse x² + 4y² - 2x - 16 + 13 = 0 is .......

(A)  The latus rectum of the ellipse is 1.

(B)  The distance between the foci is 4v3. 

(C)  The sum of the focal distances of a point P(x, y) on the ellipse is 4.

(D) The line y = 3 meets the tangents drawn at the vertices of the ellipse at points P and Q. Then PQ subtends a right angle at either of the foci. 

Answer 1

Correct option  (d)

Explanation :

The given ellipse equation is

(x - 1)² + 4(y - 2)² = -13 + 1 + 16 = 4

⇒ (x - 1)²/4 + (y - 2)² = 1

We have a² = 4,b² = 1. The eccentricity e is given by

1 = 4(1 - e²) ⇒ e = √3/2

The latus rectum is

2b²/2 = 2(1)/2 = 1

Hence, Statement (A) is true. Also, the ellipse is

X²/4 + Y²/1 = 1

where X = x - 1, Y = y - 2 , . The foci is

Hence, the distance between the foci is

(1 + √3) - (1 - √3) = 2√3

Hence, Statement (B) is not true. The sum of the focal distance is

SP + S' P = 2a = 2(2) = 4

Therefore, Statement (C) is true. The tangents at the vertices are x - 1 =  ±2 or x = 3, -1. The line y = 3 meets these tangents at (3, 3) and (−1, 3). We have P = (3, 3) and Q = (-1,3)S = (1 + √3,2) Therefore

Slope of SP x Slope of SP

Therefore, (bar)PQ subtends right angle at the focus (1, +√3,2) and also at the other focus. Hence, Statement (D) is true.