JEE Mathematics Circles MCQs Set 02

Question. The number of points on the circle \( 2x^2 + 2y^2 – 3x = 0 \) which are at a distance 2 from the point (-2, 1) is
(a) 2
(b) 0
(c) 1
(d) None of the options
Answer: (b) 0

Question. The equation of the diameter of the circle \( 3(x^2 + y^2) – 2x + 6y – 9 = 0 \) which is perpendicular to the line \( 2x + 3y = 12 \) is
(a) \( 3x – 2y = 3 \)
(b) \( 3x – 2y + 1 = 0 \)
(c) \( 3x – 2y = 0 \)
(d) None of the options
Answer: (a) \( 3x – 2y = 3 \)

Question. The equation of a circle C is \( x^2 + y^2 – 6x – 8y – 11 = 0 \). The number of real points at which the circle drawn with the points (1, 8) and (0, 0) at the ends of a diameter cuts the circle C is
(a) 0
(b) 1
(c) 2
(d) None of the options
Answer: (c) 2

Question. The equation of the circle of radius \( 2\sqrt{2} \) whose centre lies on the line \( x – y = 0 \) and which touches the line \( x + y = 4 \), and whose centre’s coordinates satisfy the inequality \( x + y > 4 \) is
(a) \( x^2 + y^2 – 8x – 8y + 24 = 0 \)
(b) \( x^2 + y^2 = 8 \)
(c) \( x^2 + y^2 – 8x + 8y = 24 \)
(d) None of the options
Answer: (a) \( x^2 + y^2 – 8x – 8y + 24 = 0 \)

Question. The equation of the chord of the circle \( x^2 + y^2 = 25 \) of length 8 that passes through the point \( (2\sqrt{3}, 2) \) and makes an acute angle with the positive direction of the x-axis is
(a) \( (4\sqrt{3} - 3\sqrt{7})x + 3y = 18 - 6\sqrt{21} \)
(b) \( (4\sqrt{3} + 3\sqrt{7})x - 3y = 18 + 6\sqrt{21} \)
(c) \( (4\sqrt{3} + 3\sqrt{7})x - 3y + 18 + 6\sqrt{21} = 0 \)
(d) None of the options
Answer: (b) \( (4\sqrt{3} + 3\sqrt{7})x - 3y = 18 + 6\sqrt{21} \)

Question. If (a, b) is a point on the chord AB of the circle, where the ends of the chord are A = (2, -3) and B = (3, 2), then
(a) \( a \in [-3, 2], b \in [2, 3] \)
(b) \( a \in [2, 3], b \in [-3, 2] \)
(c) \( a \in [-2, 2], b \in [-3, 3] \)
(d) None of the options
Answer: (b) \( a \in [2, 3], b \in [-3, 2] \)

Question. The number of points with integral coordinates that are interior to the circle \( x^2 + y^2 = 16 \) is
(a) 43
(b) 49
(c) 45
(d) 51
Answer: (c) 45

Question. The range of values of a for which the point (a, 4) is outside the circles \( x^2 + y^2 + 10x = 0 \) and \( x^2 + y^2 – 12x + 20 = 0 \) is
(a) \( (-\infty, -8) \cup (-2, 6) \cup (6, +\infty) \)
(b) \( (-8, -2) \)
(c) \( (-\infty, -8) \cup (-2, +\infty) \)
(d) None of the options
Answer: (a) \( (-\infty, -8) \cup (-2, 6) \cup (6, +\infty) \)

Question. A region in the x-y plane is bounded by the curve \( y = \sqrt{25 - x^2} \) and the line \( y = 0 \). If the point \( (a, a + 1) \) lies in the interior of the region then
(a) \( a \in (-4, 3) \)
(b) \( a \in (-\infty, -1) \cup (3, +\infty) \)
(c) \( a \in (-1, 3) \)
(d) None of the options
Answer: (c) \( a \in (-1, 3) \)

Question. If (2, 4) is a point interior to the circle \( x^2 + y^2 – 6x – 10y + \lambda = 0 \) and the circle does not cut the axes at any point then \( \lambda \) belongs to the interval
(a) \( (25, 32) \)
(b) \( (9, 32) \)
(c) \( (32, +\infty) \)
(d) None of the options
Answer: (a) \( (25, 32) \)

Question. The range of values of \( \theta \in [0, 2\pi] \) for which \( (1 + \cos \theta, \sin \theta) \) is an interior point of the circle \( x^2 + y^2 = 1 \) is
(a) \( (\pi/6, 5\pi/6) \)
(b) \( (2\pi/3, 5\pi/3) \)
(c) \( (\pi/6, 7\pi/6) \)
(d) \( (2\pi/3, 4\pi/3) \)
Answer: (d) \( (2\pi/3, 4\pi/3) \)

Question. The range of the values of \( r \) for which the point \( (-5 + \frac{r}{\sqrt{2}}, -3 + \frac{r}{\sqrt{2}}) \) is an interior point of the major segment of the circle \( x^2 + y^2 = 16 \), cut off by the line \( x + y = 2 \), is
(a) \( (-\infty, 5\sqrt{2}) \)
(b) \( (4\sqrt{2} - \sqrt{14}, 5\sqrt{2}) \)
(c) \( (4\sqrt{2} - \sqrt{14}, 4\sqrt{2} + \sqrt{14}) \)
(d) None of the options
Answer: (b) \( (4\sqrt{2} - \sqrt{14}, 5\sqrt{2}) \)

Question. There are two circles whose equations are \( x^2 + y^2 = 9 \) and \( x^2 + y^2 – 8x – 6y + n^2 = 0 \), \( n \in Z \). If the two circles have exactly two common tangents then the number of possible values of \( n \) is
(a) 2
(b) 8
(c) 9
(d) None of the options
Answer: (c) 9

Question. The number of common tangents to the circles \( x^2 + y^2 = 4 \) and \( x^2 + y^2 – 6x – 8y = 24 \) is
(a) 0
(b) 1
(c) 3
(d) 4
Answer: (b) 1

Question. The number of common tangents to the circles \( x^2 + y^2 + 2x + 8y – 23 = 0 \) and \( x^2 + y^2 – 4x – 10y + 19 = 0 \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. If the circles \( x^2 + y^2 + 2ax + c = 0 \) and \( x^2 + y^2 + 2by + c = 0 \) touch each other then
(a) \( a^{-2} + b^{-2} = c^{-1} \)
(b) \( a^{-2} + b^{-2} = c^{-2} \)
(c) \( a + b = 2c \)
(d) \( 1/a + 1/b = 2/c \)
Answer: (a) \( a^{-2} + b^{-2} = c^{-1} \)

Question. The number of common tangents to the circles one of which passes through the origin and cuts off intercepts 2 from each of the axes, and the other circle has the line segment joining the origin and the point (1, 1) as a diameter, is
(a) 0
(b) 1
(c) 3
(d) 2
Answer: (b) 1

Question. The range of values of \( \lambda \) for which the circles \( x^2 + y^2 = 4 \) and \( x^2 + y^2 - 4\lambda x + 9 = 0 \) have two common tangents, is
(a) \( \lambda \in [-13/8, 13/8] \)
(b) \( \lambda > 13/8 \) or \( \lambda < -13/8 \)
(c) \( 1 < \lambda < 13/8 \)
(d) None of the options
Answer: (b) \( \lambda > 13/8 \) or \( \lambda < -13/8 \)

Question. The number of common tangents to the circles \( x^2 + y^2 – 6x – 14y + 48 = 0 \) and \( x^2 + y^2 – 6x = 0 \) is
(a) 1
(b) 2
(c) 0
(d) 4
Answer: (d) 4

Question. Two circles have the equations \( x^2 + y^2 – 4x – 6y – 8 = 0 \) and \( x^2 + y^2 – 2x – 3 = 0 \). Then
(a) they cut each other
(b) they touch each other
(c) one circle lies inside the other
(d) one circle lies wholly outside the other
Answer: (a) they cut each other

Question. The equations of two circles are \( x^2 + y^2 – 26y + 25 = 0 \) and \( x^2 + y^2 = 25 \). Then
(a) they touch each other
(b) they cut each other orthogonally
(c) one circle is inside the other circle
(d) None of the options
Answer: (b) they cut each other orthogonally

Question. A tangent is drawn to the circle \( 2(x^2 + y^2) – 3x + 4y = 0 \) and it touches the circle at point A. The tangent passes the point P(2, 1). Then PA is equal to
(a) 4
(b) 2
(c) \( 2\sqrt{2} \)
(d) None of the options
Answer: (b) 2

Question. If the points A(1, 4) and B are symmetrical about the tangent to the circles \( x^2 + y^2 – x + y = 0 \) at the origin then coordinates of B are
(a) (1, 2)
(b) \( (\sqrt{2}, 1) \)
(c) (4, 1)
(d) None of the options
Answer: (c) (4, 1)

Question. The range of values of \( m \) for which the line \( y = mx + 2 \) cuts the circle \( x^2 + y^2 – x + y = 0 \) at the origin then coordinates of B are
(a) (1, 2)
(b) \( (\sqrt{2}, 1) \)
(c) (4, 1)
(d) None of the options
Answer: (a) (1, 2)

Question. The range of values of \( m \) for which the line \( y = mx + 2 \) cuts the circles \( x^2 + y^2 = 1 \) at distinct or coincident points is
(a) \( (-\infty, -\sqrt{3}] \cup [\sqrt{3}, +\infty) \)
(b) \( [-\sqrt{3}, \sqrt{3}] \)
(c) \( [\sqrt{3}, +\infty) \)
(d) None of the options
Answer: (b) \( [-\sqrt{3}, \sqrt{3}] \)

Question. The equation of any tangent to the circle \( x^2 + y^2 – 2x + 4y – 4 = 0 \) is
(a) \( y = m(x - 1) + 3\sqrt{1 + m^2} - 2 \)
(b) \( y = mx + 3\sqrt{1 + m^2} \)
(c) \( y = mx + 3\sqrt{1 + m^2} - 2 \)
(d) None of the options
Answer: (a) \( y = m(x - 1) + 3\sqrt{1 + m^2} - 2 \)

Question. Two tangents to the circle \( x^2 + y^2 = 4 \) at the point A and B meet at P(-4, 0). The area of the quadrilateral PAOB, where O is the origin, is
(a) 4
(b) \( 6\sqrt{2} \)
(c) \( 4\sqrt{3} \)
(d) None of the options
Answer: (c) \( 4\sqrt{3} \)

Choose the correct options. One or more options may be correct.

Question. A point \( P(\sqrt{3}, 1) \) moves on the circle \( x^2 + y^2 = 4 \) and after covering a quarter of the circle leaves it tangentially. The equation of a line along which the point moves after leaving the circle is
(a) \( y = \sqrt{3}x + 4 \)
(b) \( \sqrt{3}y = x + 4 \)
(c) \( \sqrt{3}y = x - 4 \)
(d) \( y = \sqrt{3}x - 4 \)
Answer: (b) \( \sqrt{3}y = x + 4 \) and (c) \( \sqrt{3}y = x - 4 \)

Question. The equation of a circle of radius 1 touching the circles \( x^2 + y^2 – 2|x| = 0 \) is
(a) \( x^2 + y^2 + 2\sqrt{3}x - 2 = 0 \)
(b) \( x^2 + y^2 - 2\sqrt{3}y + 2 = 0 \)
(c) \( x^2 + y^2 + 2\sqrt{3}y + 2 = 0 \)
(d) \( x^2 + y^2 + 2\sqrt{3}x + 2 = 0 \)
Answer: (b) \( x^2 + y^2 - 2\sqrt{3}y + 2 = 0 \) and (c) \( x^2 + y^2 + 2\sqrt{3}y + 2 = 0 \)

Question. The line \( 4y – 3x + \lambda = 0 \) touches the circle \( x^2 + y^2 - 4x – 8y – 5 = 0 \). The value of \( \lambda \) is
(a) 29
(b) 10
(c) -35
(d) None of the options
Answer: (a) 29 and (c) -35

Question. A circle which touches the axes, and whose centre is at distance \( 2\sqrt{2} \) from the origin, has the equation
(a) \( x^2 + y^2 - 4x + 4y + 4 = 0 \)
(b) \( x^2 + y^2 + 4x - 4y + 4 = 0 \)
(c) \( x^2 + y^2 + 4x + 4y + 4 = 0 \)
(d) None of the options
Answer: (b) \( x^2 + y^2 + 4x - 4y + 4 = 0 \) and (c) \( x^2 + y^2 + 4x + 4y + 4 = 0 \)

Question. Let the equation of a circle be \( x^2 + y^2 = a^2 \). If \( h^2 + k^2 – a^2 < 0 \) then the line \( hx + ky = a^2 \) is the
(a) polar line of the point (h, k) with respect to the circle
(b) real chord of contact of the tangents from (h, k) to the circle
(c) equation of a tangent to the circle from the point (h, k)
(d) None of the options
Answer: (a) polar line of the point (h, k) with respect to the circle

Question. For the equation \( x^2 + y^2 + 2\lambda x + 4 = 0 \) which of the following can be true?
(a) It represents a real circle for all \( \lambda \in R \).
(b) It represents a real circle for \( |\lambda| > 2 \)
(c) The radical axis of any two circles of the family is the y-axis
(d) The radical axis of any two circles of the family is the x-axis.
Answer: (b) It represents a real circle for \( |\lambda| > 2 \) and (c) The radical axis of any two circles of the family is the y-axis