Chapter 11
Circle
1. If P and Q are the points of intersection of the 7. Let C be the circle with centre at (1, 1) and radius
circles x 2 + y 2 + 3x + 7y + 2p – 5 = 0 and = 1. If T is the circle centred at (0, y), passing
x2 + y2 + 2x + 2y – p2 = 0, then there is a circle through origin and touching the circle C externally,
passing through P, Q and (1, 1) for [AIEEE-2009] then the radius of T is equal to[JEE (Main)-2014]
(1) All except one value of p
1 1
(2) All except two values of p (1) (2)
2 4
(3) Exactly one value of p
(4) All values of p 3 3
(3) (4)
2. The circle x2
+ y2
= 4x + 8y + 5 intersects the line 2 2
3x – 4y = m at two distinct points if [AIEEE-2010] 8. The number of common tangents to the circles
(1) –85 < m < –35 (2) –35 < m < 15 x2 + y2 – 4x – 6y – 12 = 0 and
x2 + y2 + 6x + 18y + 26 = 0, is[JEE (Main)-2015]
(3) 15 < m < 65 (4) 35 < m < 85
3. The equation of the circle passing through the (1) 1 (2) 2
points (1, 0) and (0, 1) and having the smallest (3) 3 (4) 4
radius is [AIEEE-2011]
9. The centres of those circles which touch the circle,
(1) x2 + y2 + 2x + 2y – 7 = 0 x2 + y2 – 8x – 8y – 4 = 0, externally and also
(2) x2 + y2 + x + y – 2 = 0 touch the x-axis, lie on [JEE (Main)-2016]
(3) x2 + y2 – 2x – 2y + 1 = 0 (1) An ellipse which is not a circle
(4) x2 + y2 – x – y = 0 (2) A hyperbola
4. The length of the diameter of the circle which (3) A parabola
touches the x-axis at the point (1, 0) and passes
through the point (2, 3) is [AIEEE-2012] (4) A circle
(1) 3/5 (2) 6/5 10. If one of the diameters of the circle, given by the
equation, x2 + y2 – 4x + 6y – 12 = 0, is a chord
(3) 5/3 (4) 10/3
of a circle S, whose centre is at (–3, 2), then the
5. The circle passing through (1, – 2) and touching radius of S is [JEE (Main)-2016]
the axis of x at (3, 0) also passes through the
point [JEE (Main)-2013] (1) 5 3
(1) (–5, 2) (2) (2, –5) (2) 5
(3) (5, –2) (4) (–2, 5) (3) 10
6. The equation of the circle passing through the foci
(4) 5 2
x2 y 2
of the ellipse 1 , and having centre at
16 9 11. The radius of a circle, having minimum area,
(0, 3) is [JEE (Main)-2013] which touches the curve y = 4 – x2 and the lines,
(1) x2 + y2 – 6y – 7 = 0 y = |x| is [JEE (Main)-2017]
(2) x2 + y2 – 6y + 7 = 0 (1) 2 2 1 (2) 4 2 1
(3) x2 + y2 – 6y – 5 = 0
(4) x2 + y2 – 6y + 5 = 0 (3) 4 2 1 (4) 2 2 1
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MATHEMATICS ARCHIVE - JEE (Main)
12. Let the orthocentre and centroid of a triangle be 18. If the area of an equilateral triangle inscribed in the
A(–3, 5) and B(3, 3) respectively. If C is the
circle, x2 + y2 + 10x + 12y + c = 0 is 27 3 sq. units
circumcentre of this triangle, then the radius of the
circle having line segment AC as diameter, is [JEE (Main)-2019]
[JEE (Main)-2018] (1) 13 (2) 25
(1) 10 (2) 2 10 (3) – 25 (4) 20
19. Two circles with equal radii are intersecting at the
5 3 5 points (0, 1) and (0, –1). The tangent at the point
(3) 3 (4)
2 2 (0, 1) to one of the circles passes through the
centre of the other circle. Then the distance
13. If the tangent at (1, 7) to the curve x2 = y – 6
between the centres of these circles is
touches the circle x2 + y2 + 16x + 12y + c = 0 then
the value of c is [JEE (Main)-2018] [JEE (Main)-2019]
(1) 195 (2) 185 (1) 1 (2) 2
(3) 85 (4) 95
(3) 2 2 (4) 2
14. Three circles of radii a, b, c (a < b < c) touch each
other externally. If they have x-axis as a common 20. A square is inscribed in the circle x2 + y2 – 6x +
tangent, then [JEE (Main)-2019] 8y – 103 = 0 with its sides parallel to the
coordinate axes. Then the distance of the vertex
(1) a, b, c are in A.P.
of this square which is nearest to the origin is
1 1 1 [JEE (Main)-2019]
(2)
a b c
(1) 6 (2) 41
(3) a, b, c are in A.P.
(3) 13 (4) 137
1 1 1 21. A circle cuts a chord of length 4a on the x-axis
(4) and passes through a point on the y-axis, distant
b a c
2b from the origin. Then the locus of the centre of
15. Equation of a common tangent to the circle, this circle, is [JEE (Main)-2019]
x2 + y2 – 6x = 0 and the parabola, y2 = 4x, is (1) A hyperbola
[JEE (Main)-2019] (2) A parabola
(1) 3y x 3 (2) 2 3 y 12 x 1 (3) An ellipse
(4) A straight line
(3) 3y 3 x 1 (4) 2 3y x 12
22. If a variable line, 3x + 4y – = 0 is such that the
16. If the circles + x2 y2
– 16x – 20y + 164 = and r2 two circles x2 + y2 – 2x – 2y + 1 = 0and x2 + y2 –
(x – 4)2 + (y – 7)2 = 36 intersect at two distinct 18x – 2y + 78 = 0 are on its opposite sides, then
points, then [JEE (Main)-2019] the set of all values of is the interval
(1) 1 < r < 11 [JEE (Main)-2019]
(2) r > 11 (1) (2, 17) (2) (12, 21)
(3) r = 11 (3) (13, 23) (4) (23, 31)
(4) 0 < r < 1 23. If a circle of radius R pases through the origin O
and intersects the coordinate axes at A and B,
17. If a circle C passing through the point (4, 0) then the locus of the foot of perpendicular from O
touches the circle x2 + y2 + 4x – 6y = 12 externally on AB is [JEE (Main)-2019]
at the point (1, –1), then the radius of C is
(1) (x 2 + y 2)2 = 4Rx 2y 2
[JEE (Main)-2019]
(2) (x 2 + y 2)2 = 4R 2x 2y 2
(1) 5 (2) 2 5
(3) (x 2 + y 2)3 = 4R 2x 2y 2
(3) 57 (4) 4 (4) (x 2 + y 2)(x + y) = R 2xy
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ARCHIVE - JEE (Main) MATHEMATICS
30. The locus of the centres of the circles, which
24. The sum of the squares of the lengths of the
touch the circle, x2 + y2 = 1 externally, also touch
chords intercepted on the circle, x2 + y2 = 16, by
the y-axis and lie in the first quadrant, is
the lines, x + y = n, nN, where N is the set of all
natural numbers, is [JEE (Main)-2019] [JEE (Main)-2019]
(1) 105 (2) 160 (1) y 1 2 x , x 0 (2) x 1 4y , y 0
(3) 320 (4) 210 (3) x 1 2y , y (4) y 1 4x , x
25. The tangent and the normal lines at the point
31. A circle touching the x-axis at (3, 0) and making
3, 1 to the circle x2 + y2 = 4 and the x-axis form an intercept of length 8 on the y-axis passes
through the point [JEE (Main)-2019]
a triangle. The area of this triangle
(in square units) is [JEE (Main)-2019] (1) (2, 3) (2) (1, 5)
(3) (3, 5) (4) (3, 10)
2 4 32. Let the tangents drawn from the origin to the
(1) (2)
3 3 circle, x2 + y2 – 8x – 4y + 16 = 0 touch it at the
points A and B. The (AB)2 is equal to
1 1 [JEE (Main)-2020]
(3) (4)
3 3
64 52
(1) (2)
26. If a tangent to the circle x2 + y2 = 1 intersects the 5 5
coordinate axes at distinct points P and Q, then
56 32
the locus of the mid-point of PQ is (3) (4)
5 5
[JEE (Main)-2019]
33. If a line, y = mx + c is a tangent to the circle,
(1) x2 + y2 – 16x2y2 =0 (x – 3)2 + y2 = 1 and it is perpendicular to a line
L 1 , where L 1 is the tangent to the circle,
(2) x2 + y2 – 2x2y2 = 0
⎛ 1 1 ⎞
(3) x2 + y2 – 4x2y2 = 0 x2 + y2 = 1 at the point ⎜ , ⎟ ; then
⎝ 2 2⎠
(4) x2 + y2 – 2xy = 0 [JEE (Main)-2020]
27. The common tangent to the circles x2 + y2 = 4 and (1) c2 + 6c + 7 = 0 (2) c2 – 7c + 6 = 0
x2 + y2 + 6x + 8y – 24 = 0 also passes through
(3) c2 + 7c + 6 = 0 (4) c2 – 6c + 7 = 0
the point [JEE (Main)-2019]
34. A circle touches the y-axis at the point (0, 4) and
(1) (–6, 4) (2) (–4, 6) passes through the point (2, 0). Which of the
(3) (4, –2) (4) (6, –2) following lines is not a tangent to this circle?
28. The line x = y touches a circle at the point [JEE (Main)-2020]
(1, 1). If the circle also passes through the point (1) 3x – 4y – 24 = 0 (2) 4x – 3y + 17 = 0
(1, –3), then its radius is [JEE (Main)-2019] (3) 4x + 3y – 8 = 0 (4) 3x + 4y – 6 = 0
(1) 3 2 (2) 2 35. Let the latus ractum of the parabola y2 = 4x be the
common chord to the circles C1 and C2 each of
(3) 2 2 (4) 3 them having radius 2 5 . Then, the distance
29. If the circles + x2 y2
+ 5Kx + 2y + K = 0 and between the centres of the circles C1 and C2 is
2(x2 + y2) + 2Kx + 3y – 1 = 0, (K R), intersect [JEE (Main)-2020]
at the points P and Q, then the line 4x + 5y – K (1) 8 (2) 12
= 0 passes through P and Q, for
(3) 8 5 (4) 4 5
[JEE (Main)-2019]
36. The circle passing through the intersection of the
(1) Exactly one value of K circles, x2 + y2 – 6x = 0 and x2 + y2 – 4y = 0,
(2) Infinitely many values of K having its centre on the line, 2x – 3y + 12 = 0, also
passes through the point [JEE (Main)-2020]
(3) Exactly two values of K
(1) (–3, 6) (2) (–1, 3)
(4) No value of K (3) (–3, 1) (4) (1, –3)
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MATHEMATICS ARCHIVE - JEE (Main)
37. If the co-ordinates of two points A and B are 39. The centre of the circle passing through the point
(0, 1) and touching the parabola y = x2 at the point
( 7, 0) and ( 7, 0) respectively and P is any (2, 4) is [JEE (Main)-2020]
point on the conic, 9x 2 + 16y 2 = 144, then
⎛ 53 16 ⎞ ⎛ 6 53 ⎞
PA + PB is equal to [JEE (Main)-2020] (1) ⎜ , ⎟ (2) ⎜ , ⎟
⎝ 10 5 ⎠ ⎝ 5 10 ⎠
(1) 9 (2) 16
⎛ 16 53 ⎞ ⎛ 3 16 ⎞
(3) 6 (4) 8 (3) ⎜ , ⎟ (4) ⎜ , ⎟
⎝ 5 10 ⎠ ⎝ 10 5 ⎠
38. If the length of the chord of the circle,
40. The number of integral values of k for which the
x2 + y2 = r2 (r > 0) along the line, y – 2x = 3 is r, line, 3x + 4y = k intersects the circle,
then r2 is equal to [JEE (Main)-2020] x2 + y2 – 2x – 4y + 4 = 0 at two distinct points
is ______. [JEE (Main)-2020]
9 24
(1) (2) 41. The diameter of the circle, whose centre lies on
5 5
the line x + y = 2 in the first quadrant and which
touches both the lines x = 3 and y = 2, is _____.
12
(3) (4) 12 [JEE (Main)-2020]
5
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