Class 11 Mathematics Conic Sections MCQs

Question. If the tangent at (1, 7) to the curve x² = y - 6 touches the circle x² + y² +16x +12y + c = 0 then the value of c is :
(a) 185
(b) 85
(c) 95
(d) 195
Answer : C

Question. If a circle C, whose radius is 3, touches externally the circle, x² + y² + 2x – 4y – 4 = 0 at the point (2, 2), then the length of the intercept cut by this circle c, on the x-axis is equal to
(a) √5
(b) 2√3
(c) 3√2
(d) 2√5
Answer : D

Question. If one of the diameters of the circle, given by the equation, x² + y² – 4x + 6y – 12 = 0, is a chord of a circle S, whose centre is at (–3, 2), then the radius of S is: 
(a) 5
(b) 10
(c) 5√2
(d) 5√3
Answer : D

Question. The two ad acent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60 . If the area of the quadrilateral is 4√3 , then the perimeter of the quadrilateral is :
(a) 12.5
(b) 13.2
(c) 12
(d) 13
Answer : C

Question. The tangent to the circle C₁ : x² + y² – 2x – 1 = 0 at the point (2, 1) cuts off a chord of length 4 from a circle C₂ whose centre is (3, – 2). The radius of C₂ is
(a) √6
(b) 2
(c) √2
(d) 3
Answer : A

Question. The radius of a circle, having minimum area, which touches the curve y = 4 – x² and the lines, y = |x| is :
(a) 4(√2 +1)
(b) 2(√2 +1)
(c) 2(√2 -1)
(d) 4(√2 -1)
Answer : C

Question. If the point P on the curve, 4x² + 5y² = 20 is farthest from the point Q(0, – 4), then PQ2 is equals to :
(a) 36
(b) 48
(c) 21
(d) 29
Answer : A

Question. Statement 1: y = mx - 1/m is always a tangent to the parabola, y² = – 4x for all non- ero values of m.
Statement 2: Every tangent to the parabola, y² = – 4x will meet its axis at a point whose abscissa is non-negative.
(a) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
(b) Statement 1 is false, Statement 2 is true.
(c) Statement 1 is true, Statement 2 is false.
(d) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
Answer : D

Question. The shortest distance between line y – x =1 and curve x = y² is
(a) 3√2/8
(b) 8/3√2
(c) 4/√3
(d) √3/4
Answer : A

Question. The equation of a tangent to the parabola y² = 8x is y = x + 2. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is
(a) (2, 4)
(b) (–2, 0)
(c) (–1, 1)
(d) (0, 2)
Answer : B

Question. Let P be the point ( 1, 0 ) and Q a point on the locus 2 y =8x . The locus of mid point of PQ is
(a) y² – 4x + 2 = 0
(b) y² + 4x + 2 = 0
(c) x² + 4y + 2 = 0
(d) x² – 4y + 2 = 0
Answer : A

Question. A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is
(a) an ellipse
(b) a circle
(c) a hyperbola
(d) a parabola
Answer : D

Question. If a ≠ 0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection of the parabolas y² = 4ax and x² = 4ay, then
(a) d² + (3b - 2c)² = 0
(b) d²+ (3b + 2c)² = 0
(c) d² + (2b - 3c)² = 0
(d) d² + (2b + 3c)² = 0
Answer : D

Question. Two common tangents to the circle x² + y² = 2a² and parabola y² = 8ax are
(a) x = ±( y + 2a)
(b) y = ±(x + 2a)
(c) x = ±( y + a)
(d) y = ±(x + a)
Answer : B

Question. Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, x²/4 + y²/2 = 1 from any of its foci?
(a) (-2,√3)
(b) (-1, √2)
(c) (-1, 3)
(d) (1, 2√)
Answer : C

Question. If the normal at an end of a latus rectum of an ellipse passes through an extermity of the minor axis, then the eccentricity e of the ellipse satisfies: 
(a) e⁴ + 2e² – 1 = 0
(b) e² + e – 1 = 0
(c) e⁴ + e² – 1 = 0
(d) e² + 2e – 1 = 0
Answer : C

Question. If the co-ordinates of two points A and B are (√ 7,0) and (- √7,0) respectively and P is any point on the conic, 9x² + 16y² = 144, then PA + PB is equal to :
(a) 16
(b) 8
(c) 6
(d) 9
Answer : B

Question. The point of intersection of the normals to the parabola y² = 4x at the ends of its latus rectum is :
(a) (0, 2)
(b) (3, 0)
(c) (0, 3)
(d) (2, 0)
Answer : B

Question. Let x²/a² + y²/b² = > be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, Φ (t) = 5/12 + t - t² then a² + b² is equal to:
(a) 145
(b) 116
(c) 126
(d) 135
Answer : C

Question. Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is 1/2. If P(1, b), b > 0 is a point on this ellipse, then the equation of the normal to it at P is :
(a) 4x – 3y = 2
(b) 8x – 2y = 5
(c) 7x – 4y = 1
(d) 4x – 2y = 1
Answer : D

Question. A circle passes through the points (2, 3) and (4, 5). If its centre lies on the line, y – 4x + 3 = 0, then its radius is equal to 
(a) √5
(b) 1
(c) √2
(d) 2
Answer : C

Question. Tangents drawn from the point (– 8, 0) to the parabola y² = 8x touch the parabola at P and Q. If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to 
(a) 48
(b) 32
(c) 24
(d) 64
Answer : A

Question. Let P be the point on the parabola, y² = 8x which is at a minimum distance from the centre C of the circle, x² + (y + 6)² = 1. Then the equation of the circle, passing through C and having its centre at P is: ]
(a) x² + y² - x/4 + 2y - 24 = 0
(b) x² + y² – 4x + 9y + 18 = 0
(c) x² + y² – 4x + 8y + 12 = 0
(d) x² + y² – x + 4y – 12 = 0
Answer : C

Question. Given : A circle, 2x² + 2y² = 5 and a parabola, y² = 4√5x.
Statement-1 : An equation of a common tangent to these curves is y = x + √5 .
Statement-2 : If the line, y = mx + √5/m (m ¹ 0) is their
common tangent, then m satisfies m4 – 3m² + 2 = 0.
(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 true; Statement-2 is false.
(d) Statement-1 is false; Statement-2 is true.
Answer : B

Question. A parabola has the origin as its focus and the line x = 2 as the directrix. Then the vertex of the parabola is at
(a) (0, 2)
(b) (1, 0)
(c) (0, 1)
(d) (2, 0)
Answer : B

Question. A line drawn through the point P(4, 7) cuts the circle x² + y² = 9 at the points A and B. Then PA·PB is equal to :
(a) 53
(b) 56
(c) 74
(d) 65
Answer : B

Question. Two parabolas with a common vertex and with axes along x-axis and y-axis, respectively, intersect each other in the first quadrant. if the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is?
(a) 3 (x + y) + 4 = 0
(b) 8 (2x + y) + 3 = 0
(c) 4 (x + y) + 3 = 0
(d) x + 2y + 3 = 0
Answer : C

Question. Equation of the tangent to the circle, at the point (1, –1) whose centre is the point of intersection of the straight lines x – y = 1 and 2x + y = 3 is : 
(a) x + 4y + 3 = 0
(b) 3x – y – 4 = 0
(c) x – 3y – 4 = 0
(d) 4x + y – 3 = 0
Answer : A

Question. The equation Im (iz - 2 / z - 1) + 1 = 0, z ∈ C, z ≠ i represents a part of a circle having radius equal to :
(a) 2
(b) 1
(c) 3/4
(d) 1/2
Answer : C

Question. The number of common tangents to the circles x² + y² – 4x – 6x – 12 = 0 and x² + y² + 6x + 18y + 26 = 0, is:
(a) 3
(b) 4
(c) 1
(d) 2
Answer : A

Question. If a point P has co–ordinates (0, –2) and Q is any point on the circle, x² + y² – 5x – y + 5 = 0, then the maximum value of (PQ)² is :
(a) 25 + √6 / 2
(b) 14 + 5√6
(c) 47 + 10√6 / 2
(d) 8 + 5 √3
Answer : B

Question. If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the centre and subtend angles cos^-1 (1/7) and sec^–1 (7) at the centre respectively, then the distance between these chords, is :
(a) 4/√7
(b) 8/√7
(c) 8/7
(d) 16/7
Answer : B

Question. Let z ∈ C, the set of complex numbers. Then the equation, 2|z+3i| – |z – i| = 0 represents :
(a) a circle with radius 8/3.
(b) a circle with diameter 10/3.
(c) an ellipse with length of ma or axis 16/3.
(d) an ellipse with length of minor axis 16/9
Answer : A

Question. Locus of the image of the point (2, 3) in the line (2x – 3y + 4) + k (x – 2y + 3) = 0, k ∈ R, is a :
(a) circle of radius √2.
(b) circle of radius √3.
(c) straight line parallel to x-axis
(d) straight line parallel to y-axis
Answer : A

Question. A circle passes through (–2, 4) and touches the y-axis at (0, 2). Which one of the following equations can represent a diameter of this circle ?
(a) 2x – 3y + 10 = 0
(b) 3x + 4y – 3 = 0
(c) 4x + 5y – 6 = 0
(d) 5x + 2y + 4 = 0
Answer : A