Q81.A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 1/2. Then the length of the semi-major axis is (1) 8 (2) 2 3 3 (3) 4 (4) 5 3 3

Q69.Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (−3, 1) and has eccentricity is √25 (1) 5x2 + 3y2 −48 = 0 (2) 3x2 + 5y2 −15 = 0 (3) 5x2 + 3y2 −32 = 0 (4) 3x2 + 5y2 −32 = 0 Q70.$$ \lim _{x \rightarrow 2}\left(\frac{\sqrt{1-\cos \{2(x-2)\}}}{x-2}\right) (1) equals √2 (2) equals −√2 (3) equals 1 (4) does not exist √2

Q69.If P1 and P2 are two points on the ellipse x24 + y2 = 1 at which the tangents are parallel to the chord joining the points (0, 1) and (2, 0), then the distance between P1 and P2 is (1) 2√2 (2) √5 (3) 2√3 (4) √10

Q73.If a and c are positive real numbers and the ellipse x2 + y2 = 1 has four distinct points ir common with the 4c2 c2 circle x2 + y2 = 9a2 , then (1) 9ac −9a2 −2c2 < 0 (2) 6ac + 9a2 −2c2 < 0 (3) 9ac −9a2 −2c2 > 0 (4) 6ac + 9a2 −2c2 > 0