Class 11 Mathematics Circles MCQs Set 01

EQUATION OF CIRCLE, CENTRE-RADIUS FORM

Question. The equation of the circle with radius 3 and centre as the point of intersection of the lines \( 2x + 3y = 5, 2x - y = 1 \) is
(a) \( x^2 + y^2 = 9 \)
(b) \( x^2 + y^2 - 2x - 2y - 7 = 0 \)
(c) \( x^2 + y^2 - 2x - 2y + 7 = 0 \)
(d) \( x^2 + y^2 + 9 = 0 \)
Answer: (b) \( x^2 + y^2 - 2x - 2y - 7 = 0 \)

Question. The circle concentric with \( x^2 + y^2 + 4x + 6y + 3 = 0 \) and radius 2 is
(a) \( x^2 + y^2 + 4x + 6y - 9 = 0 \)
(b) \( x^2 + y^2 + 4x + 6y + 9 = 0 \)
(c) \( x^2 + y^2 - 4x - 6y + 9 = 0 \)
(d) \( x^2 + y^2 = 4 \)
Answer: (b) \( x^2 + y^2 + 4x + 6y + 9 = 0 \)

Question. Find the equation of the circle passing through (-2, 14) and concentric with the circle \( x^2 + y^2 - 6x - 4y - 12 = 0 \).
(a) \( x^2 + y^2 - 6x - 4y - 156 = 0 \)
(b) \( x^2 + y^2 - 6x + 4y - 156 = 0 \)
(c) \( x^2 + y^2 - 6x + 4y + 156 = 0 \)
(d) \( x^2 + y^2 + 6x + 4y + 156 = 0 \)
Answer: (a) \( x^2 + y^2 - 6x - 4y - 156 = 0 \)

Question. If the centroid of an equilateral triangle is (1,1) and one of its vertices is (-1,2) then, equation of its circum circle is
(a) \( x^2 + y^2 - 2x - 2y - 3 = 0 \)
(b) \( x^2 + y^2 + 2x - 2y - 3 = 0 \)
(c) \( x^2 + y^2 - 4x - 6y + 9 = 0 \)
(d) \( x^2 + y^2 + x - y + 5 = 0 \)
Answer: (a) \( x^2 + y^2 - 2x - 2y - 3 = 0 \)

Question. For the circle \( ax^2 + y^2 + bx + dy + 2 = 0 \) center is (1,2) then 2b+3d=
(a) -16
(b) 16
(c) 8
(d) -8
Answer: (a) -16

Question. The radius of the circle passing through (6 , 2) and the equations of two normals for the circle are x + y = 6 and x+2y = 4 is
(a) \( \sqrt{5} \)
(b) \( 2\sqrt{5} \)
(c) \( 3\sqrt{5} \)
(d) \( 4\sqrt{5} \)
Answer: (b) \( 2\sqrt{5} \)

Question. Origin is the centre of circle passing through the vertices of an equilateral triangle whose median is of length 3a then equation of the circle is
(a) \( x^2 + y^2 = a^2 \)
(b) \( x^2 + y^2 = 2a^2 \)
(c) \( x^2 + y^2 = 3a^2 \)
(d) \( x^2 + y^2 = 4a^2 \)
Answer: (d) \( x^2 + y^2 = 4a^2 \)

Question. The diameters of a circle are along 2x+y-7=0 and x+3y-11=0. Then the equation of this circle which also passes through (5 , 7) is
(a) \( x^2 + y^2 - 4x - 6y - 16 = 0 \)
(b) \( x^2 + y^2 - 4x - 6y - 20 = 0 \)
(c) \( x^2 + y^2 - 4x - 6y - 12 = 0 \)
(d) \( x^2 + y^2 + 4x + 6y - 12 = 0 \)
Answer: (c) \( x^2 + y^2 - 4x - 6y - 12 = 0 \)

Question. If the two circles \( x^2 + y^2 + 2gx + c = 0 \) and \( x^2 + y^2 - 2fy - c = 0 \) have equal radius then locus of (g , f) is
(a) \( x^2 + y^2 = c^2 \)
(b) \( x^2 - y^2 = 2c \)
(c) \( x^2 - y^2 = 4 \)
(d) \( x^2 + y^2 = 2c^2 \)
Answer: (b) \( x^2 - y^2 = 2c \)

Question. Centre and radius of the circle with segment of the line x+y=1 cut off by coordinate axes as diameter is
(a) \( (\frac{1}{2}, \frac{1}{2}), \frac{1}{\sqrt{2}} \)
(b) \( (-\frac{1}{2}, -\frac{1}{2}), \frac{1}{\sqrt{2}} \)
(c) \( (\frac{1}{2}, -\frac{1}{2}), \frac{1}{\sqrt{2}} \)
(d) \( (-\frac{1}{2}, \frac{1}{2}), \frac{1}{\sqrt{2}} \)
Answer: (a) \( (\frac{1}{2}, \frac{1}{2}), \frac{1}{\sqrt{2}} \)

SHORTEST DISTANCE AND LONGEST DISTANCE

Question. The shortest distance from (-2, 14) to the circle \( x^2 + y^2 - 6x - 4y - 12 = 0 \) is
(a) 4
(b) 6
(c) 8
(d) 10
Answer: (c) 8

Question. The least distance of the line 8x-4y+73=0 from the circle \( 16x^2 + 16y^2 + 48x - 8y - 43 = 0 \) is
(a) \( \sqrt{5}/2 \)
(b) \( 2\sqrt{5} \)
(c) \( 3\sqrt{5} \)
(d) \( 4\sqrt{5} \)
Answer: (b) \( 2\sqrt{5} \)

CONCYCLIC POINTS, CIRCUMSCRIBING CIRCLES

Question. If the points (0,0), (2,0), (0,-2), and (k,-2) are concyclic then k=
(a) 2
(b) -2
(c) 0
(d) 1
Answer: (a) 2

Question. If a circle is inscribed in a square of side 10, so that the circle touches the four sides of the square internally then radius of the circle is
(a) 10
(b) \( 5\sqrt{2} \)
(c) \( 10\sqrt{2} \)
(d) 5
Answer: (d) 5

Question. The centre of the circle circumscribing the square whose three sides are \( 3x + y = 22, x - 3y = 14 \) and \( 3x + y = 62 \) is:
(a) \( (\frac{3}{2}, \frac{27}{2}) \)
(b) \( (\frac{27}{2}, \frac{3}{2}) \)
(c) (27, 3)
(d) \( (1, \frac{2}{3}) \)
Answer: (b) \( (\frac{27}{2}, \frac{3}{2}) \)

Question. If the points (2,0), (0,1), (4,0) and (0,a) are concyclic then a=
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (d) 8

Question. Centre of the circle inscribed in a rectangle formed by the lines \( x^2 - 8x + 12 = 0 \) and \( y^2 - 14y + 40 = 0 \) is
(a) (4, 7)
(b) (7, 4)
(c) (9, 4)
(d) (4, 9)
Answer: (a) (4, 7)

POWER OF A POINT, POSITION OF A POINT, CHORD

Question. The power of (1,1) with respect to the circle \( x^2 + y^2 - 4x + 3y + k = 0 \) is 3 then k is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. The equation of the circle with centre (3,2) and the power of (1,-2) w.r.t the circle \( x^2 + y^2 = 1 \) as radius is
(a) \( x^2 + y^2 - 6x - 4y - 3 = 0 \)
(b) \( x^2 + y^2 - 3x - 2y - 3 = 0 \)
(c) \( x^2 + y^2 + 6x + 4y - 3 = 0 \)
(d) \( x^2 + y^2 - 6x - 4y + 3 = 0 \)
Answer: (a) \( x^2 + y^2 - 6x - 4y - 3 = 0 \)

Question. If a line is drawn through a point A(3,4) to cut the circle \( x^2 + y^2 = 4 \) at P and Q then \( AP \cdot AQ = \)
(a) 15
(b) 17
(c) 21
(d) 25
Answer: (c) 21

Question. A chord of length 24 units is at a distance of 5 units from the center of a circle then its radius is
(a) 5
(b) 12
(c) 13
(d) 10
Answer: (c) 13

Question. If the line \( 3x - 4y = \lambda \) cuts the circle \( x^2 + y^2 - 4x - 8y - 5 = 0 \) in two points then limits of \( \lambda \) are 
(a) [-35,15]
(b) (-35,15)
(c) (-35,10)
(d) (-35,15]
Answer: (b) (-35,15)

TANGENT AND NORMAL

Question. The length of the tangent from (1,1) to the circle \( 2x^2 + 2y^2 + 5x + 3y + 1 = 0 \) is
(a) \( \sqrt{13/2} \)
(b) 3
(c) 2
(d) 1
Answer: (d) 1

Question. The locus of the point from which the length of the tangent to the circle \( x^2 + y^2 - 2x - 4y + 4 = 0 \) is 3 units is
(a) \( x^2 + y^2 - 2x - 4y - 9 = 0 \)
(b) \( x^2 + y^2 - 2x - 4y - 4 = 0 \)
(c) \( x^2 + y^2 - 2x - 4y - 3 = 0 \)
(d) \( x^2 + y^2 - 2x - 4y - 5 = 0 \)
Answer: (d) \( x^2 + y^2 - 2x - 4y - 5 = 0 \)

Question. The tangent to the circle \( x^2 + y^2 - 4x + 2y + k = 0 \) at (1,1) is \( x - y + 2y + 1 = 0 \) then k=
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (d) 2

Question. The equations of the tangents to the circle \( x^2 + y^2 = 25 \) with slope 2 is
(a) \( y = 2x \pm 5 \)
(b) \( y = 2x \pm 2\sqrt{5} \)
(c) \( y = 2x \pm 3\sqrt{5} \)
(d) \( y = 2x \pm 5\sqrt{5} \)
Answer: (d) \( y = 2x \pm 5\sqrt{5} \)

Question. The line \( 4y - 3x + \lambda = 0 \) touches the circle \( x^2 + y^2 - 4x - 8y - 5 = 0 \) then \( \lambda = \)
(a) 29
(b) 10
(c) -35
(d) 35
Answer: (c) -35

Question. The circle to which two tangents can be drawn from origin is
(a) \( x^2 + y^2 - 8x - 4y - 3 = 0 \)
(b) \( x^2 + y^2 + 4x + 2y + 2 = 0 \)
(c) \( x^2 + y^2 - 8x + 6y + 1 = 0 \)
(d) both (b) & (c)
Answer: (d) both (b) & (c)

Question. The normal at (1,1) to the circle \( x^2 + y^2 - 4x + 6y - 4 = 0 \) is
(a) 4x+3y=7
(b) 4x+y=5
(c) x+y=2
(d) 4x-y=5
Answer: (b) 4x+y=5

Question. Slopes of tangents through (7, 1) to the circle \( x^2 + y^2 = 25 \) satisfy the equation
(a) \( 12m^2 + 7m + 12 = 0 \)
(b) \( 12m^2 - 7m + 12 = 0 \)
(c) \( 12m^2 + 7m - 12 = 0 \)
(d) \( 12m^2 - 7m - 12 = 0 \)
Answer: (d) \( 12m^2 - 7m - 12 = 0 \)

Question. Angle between tangents drawn from a point P to circle \( x^2 + y^2 - 4x - 8y + 8 = 0 \) is \( 60^\circ \) then length of chord of contact of P is
(a) 6
(b) 4
(c) 2
(d) 3
Answer: (a) 6

Question. Locus of the point of intersection of perpendicular tangents to the circle \( x^2 + y^2 = 10 \) is
(a) \( x^2 + y^2 = 5 \)
(b) \( x^2 + y^2 = 20 \)
(c) \( x^2 + y^2 = 10 \)
(d) \( x^2 + y^2 = 100 \)
Answer: (b) \( x^2 + y^2 = 20 \)

Question. Locus of the point of intersection of perpendicular tangents drawn one to each of the circles \( x^2 + y^2 = 8 \) and \( x^2 + y^2 = 12 \) is
(a) \( x^2 + y^2 = 4 \)
(b) \( x^2 + y^2 = 20 \)
(c) \( x^2 + y^2 = 208 \)
(d) \( x^2 + y^2 = 16 \)
Answer: (b) \( x^2 + y^2 = 20 \)

Question. The locus of the point of intersection of perpendicular tangents drawn to each of circles \( x^2 + y^2 = 16 \) and \( x^2 + y^2 = 9 \) is a circle whose diameter is
(a) 5
(b) \( \sqrt{7} \)
(c) \( 2\sqrt{7} \)
(d) 10
Answer: (d) 10

Question. The equation of the circle with centre at (4,3) and touching the line 5x-12y-10=0 is
(a) \( x^2 + y^2 - 4x - 6y + 4 = 0 \)
(b) \( x^2 + y^2 + 6x - 8y + 16 = 0 \)
(c) \( x^2 + y^2 - 8x - 6y + 21 = 0 \)
(d) \( x^2 + y^2 - 24x - 10y + 144 = 0 \)
Answer: (c) \( x^2 + y^2 - 8x - 6y + 21 = 0 \)

Question. The locus of points from which lengths of tangents to the two circles \( x^2 + y^2 + 4x + 3 = 0 \), \( x^2 + y^2 - 6x + 5 = 0 \) are in the ratio 2 : 3 is a circle with centre
(a) (6,0)
(b) (-6,0)
(c) (0,6)
(d) (0,-6)
Answer: (b) (-6,0)

CIRCLES TOUCHING COORDINATE AXES AND INTERCEPTS ON AXES

Question. The circle with centre (4,-1) and touching x-axis is
(a) \( x^2 + y^2 - 8x + 2y + 16 = 0 \)
(b) \( x^2 + y^2 + 18x - 2y - 16 = 0 \)
(c) \( x^2 + y^2 - 4x + y + 4 = 0 \)
(d) \( x^2 + y^2 + 14x - y + 4 = 0 \)
Answer: (a) \( x^2 + y^2 - 8x + 2y + 16 = 0 \)

Question. If the line hx + ky = 1/a touches the circle \( x^2 + y^2 = a^2 \) then the locus of (h,k) is circle of radius
(a) 1/a
(b) \( a^2 \)
(c) a
(d) \( 1/a^2 \)
Answer: (d) \( 1/a^2 \)

Question. The circle \( x^2 + y^2 - 2ax - 2ay + a^2 = 0 \) touches axes of co-ordinates at
(a) (a,a), (0,0)
(b) (a,0), (0,0)
(c) (a,0), (0,a)
(d) (0,a), (1,a)
Answer: (c) (a,0), (0,a)

Question. The y-intercept of the circle \( x^2 + y^2 + 4x + 8y - 5 = 0 \) is
(a) \( 2\sqrt{21} \)
(b) \( 2\sqrt{19} \)
(c) 6
(d) 12
Answer: (a) \( 2\sqrt{21} \)