JEE Mathematics Ellipse MCQs Set B

Question.  If x cos α + y sin α = P is a tangent to the ellipse x²/a² + y²b² =1 then-
(a) a² cos2 α + b² sin² α = P²
(b) a² sin2 α + b² cos² α = P²
(c) Both
(d) None of these

Answer: A

Question. The equation of tangents to the ellipse 9x² + 16y² = 144 which pass through the point (2, 3) is
(a) y = 3 ; x + y = 5
(b) x + y = 2
(c) y = 3
(d) x – y = 3

Answer: A

Question. Chords of an ellipse are drawn through the positive end of the minor axes. Then their mid point lies on -
(a) an ellipse
(b) a circle
(c) a parabola
(d) a hyperbola

Answer: A

Question. The line x = at² meets the ellipse x²/a² + y²/b² = 1 in the real points if -
(a) | t | ≤ 1
(b) | t | < 2
(c) | t | >1
(d) None of these

Answer: A

Question. The eccentric angles of the extremities of latus rectum of the ellipse x²/a² + y²/b² =1 is
(a) tan^-1(±ae/b)
(b) tan^-1(±be/a)
(c) tan^-1(±b/ae)
(d) None of these

Answer: A

Question. Product of the perpendiculars from the foci upon any tangent to the ellipse x²/a²  +  y²/b²=1 is
(a) b²
(b) a
(c) b
(d) a²

Answer: A

Question. The equation of the ellipse which passes through origin and has its foci at the points (1, 0) and (3, 0) is-
(a) 3x² + 4y² = 12x
(b) 3x² + y² = 12x
(c) 3x² + 4y² = x
(d) x² + 4y² = 12x

Answer: A

Question. The point of the intersection of the tangent at the two point on the ellipse x²/a² + y²/b² =1 whose eccentricity differ by a right angle lies on the ellipse is
(a) x²/a² + y²/b² =2
(b) x²/a² + y²/b² = -2
(c) x/a + y/b = 1
(d) None of these

Answer: A

Question. Find the equation of the ellipse whose eccentricity is 1/2, the focus is (–1, 1) and the directrix is x – y + 3 = 0.
(a) 7x² + 7y²+ 10x – 10y + 2xy + 7 = 0
(b) 7x² + 7y²– 10x + 10y + 2xy + 7 = 0
(c) 5x² + 7y²+ 10x – 12y + 2xy + 7 = 0
(d) x² + 5y²+ 10x + 10y + 2xy + 7 = 0

Answer: A

Question. Centre, length of the axes and eccentricity of the ellipse 2x²+3y²–4x–12y+13 = 0 are
(a) (1, 2) ; √2 ; 2/√3; 1/√ 3
(b) (2, 2) ; √3 ;2 / √3; 2/ √3
(c) Both
(d) None of these

Answer: A

Question. The radius of the circle passing through the foci of the ellipse x²/16  +  y²/9 = 1, and having its centre (0,3) is-
(a) 4
(b) 3
(c) 7/2
(d) None of these

Answer: A

Question. The eccentricity of the ellipse represented by the equation 25x² + 16y² – 150 x – 175 = 0 is-
(a) 3/5
(b) None of these
(c) 2/5
(d) 4/5

Answer: A

Question. Consider an ellipse x²/a² + y²/b²= 1, centred at point O and having AB and CD as its major and minor axes respectively. If S₁ be one of the foci of the ellipse, radius of incircle of triangle OCS₁, be 1 unit and OS₁ = 6 units, then If S be the director circle of ellipse then the equation of director circle of S is –
(a) x² + y² =√ 48.5
(b) x² + y² = 97
(c) x² + y² = 48.5
(d) None of these

Answer: A

Question. Statement 1 : Ellipse x²/25 + y²/16 = 1and hyperbola 12x² – 4y² = 27 intersect each other at right angle.
Statement 2 : Whenever focal conics intersect, they intersect each other orthogonally.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(c) Statement -1 is False, Statement-2 is True.
(d) Statement -1 is True, Statement-2 is False.

Answer: A

Question. Statement-1 : Locus of centre of a variable circle touching two circles (x – 1)² + (y – 2)² = 25 and (x – 2)² + (y – 1)² = 16 is an ellipse.
Statement-2 : If circle S₁ = 0 lies completely inside the circle S₂ = 0 then locus of centre of a variable circle S = 0 which touches both the circles is an ellipse
(a) Statement -1 is False, Statement-2 is True.
(b) Statement -1 is True, Statement-2 is False.
(c) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(d) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Answer: A