Circle PYQs - Last 10 Years

The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), (1, 1) is 18 units, is a circle of diameter d. Then d2 is equal to _____________.

If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point (\(-\)30, 0) and is tangent to the parabola y2 = 30x, then the length of this chord is :

A circle C touches the line x = 2y at the point (2, 1) and intersects the circle C1 : x2 + y2 + 2y \(-\) 5 = 0 at two points P and Q such that PQ is a diameter of C1. Then the diameter of C is :

If one of the diameters of the circle x2 + y2 - 2x - 6y + 6 = 0 is a chord of another circle 'C',
whose center is at (2, 1), then its radius is ________.

If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle (x \(-\) 2)2 + (y \(-\) 3)2 = 25 at the point (5, 7) is A, then 24A is equal to _________.

Let a point P be such that its distance from the point (5, 0) is thrice the distance of P from the point (\(-\)5, 0). If the locus of the point P is a circle of radius r, then 4r2 is equal to ________

Let the circle S : 36x2 + 36y2 \(-\) 108x + 120y + C = 0 be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, x \(-\) 2y = 4 and 2x \(-\) y = 5 lies inside the circle S, then :

Let r1 and r2 be the radii of the largest and smallest circles, respectively, which pass through the point (\(-\)4, 1) and having their centres on the circumference of the circle x2 + y2 + 2x + 4y \(-\) 4 = 0. If $${{{r_1}} \over {{r_2}}} =...

For the four circles M, N, O and P, following four equations are given :Circle M : x2 + y2 = 1Circle N : x2 + y2 \(-\) 2x = 0Circle O : x2 + y2 \(-\) 2x \(-\) 2y + 1 = 0Circle P : x2 + y2 \(-\) 2y = 0If the centre of circle M is joined with...

Choose the correct statement about two circles whose equations are given below :x2 + y2 \(-\) 10x \(-\) 10y + 41 = 0x2 + y2 \(-\) 22x \(-\) 10y + 137 = 0

Let S1 : x2 + y2 = 9 and S2 : (x \(-\) 2)2 + y2 = 1. Then the locus of center of a variable circle S which touches S1 internally and S2 externally always passes through the points :

Choose the incorrect statement about the two circles whose equations are given below :x2 + y2 \(-\) 10x \(-\) 10y + 41 = 0 and x2 + y2 \(-\) 16x \(-\) 10y + 80 = 0

The minimum distance between any two points P1 and P2 while considering point P1 on one circle and point P2 on the other circle for the given circles' equationsx2 + y2 \(-\) 10x \(-\) 10y + 41 = 0x2 + y2 \(-\) 24x \(-\) 10y + 160 = 0 is ___...

The line 2x \(-\) y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x \(-\) 2y = 4. Then, the radius of the circle is :

Let the tangent to the circle x2 + y2 = 25 at the point R(3, 4) meet x-axis and y-axis at points P and Q, respectively. If r is the radius of the circle passing through the origin O and having centre at the incentre of the triangle OPQ, the...

Two tangents are drawn from a point P to the circle x2 + y2 \(-\) 2x \(-\) 4y + 4 = 0, such that the angle between these tangents is \({\tan ^{ - 1}}\left( {{{12} \over 5}} \right)\), where \({\tan ^{ - 1}}\left( {{{12} \over 5}} \right)\) ...

Let the lengths of intercepts on x-axis and y-axis made by the circle x2 + y2 + ax + 2ay + c = 0, (a < 0) be 2\({\sqrt 2 }\) and 2\({\sqrt 5 }\), respectively. Then the shortest distance from origin to a tangent to this circle which is perp...

A circle touches the y-axis at the point (0, 4)
and passes through the point (2, 0). Which of
the following lines is not a tangent to this circle?

If the curves, x2 – 6x + y2 + 8 = 0 and
x2 – 8y + y2 + 16 – k = 0, (k > 0) touch each other
at a point, then the largest value of k is ______.

If a line, y = mx + c is a tangent to the circle,
(x – 3)2 + y2 = 1 and it is perpendicular to a line L1, where L1 is the tangent to the circle, x2 + y2 = 1 at the point \(\left( {{1 \over {\sqrt 2 }},{1 \over {\sqrt 2 }}} \right)\), then :

Let the tangents drawn from the origin to the circle, x2
+ y2
- 8x - 4y + 16 = 0 touch it at the
points A and B. The (AB)2
is equal to :

If the length of the chord of the circle,
x2 + y2 = r2
(r > 0) along the line, y – 2x = 3 is r,
then r2
is equal to :

Let PQ be a diameter of the circle x2 + y2 = 9. If \(\alpha\) and \(\beta\) are the lengths of the perpendiculars from P and Q on the straight line, x + y = 2 respectively, then the maximum value of \(\alpha\beta\) is _____.

The circle passing through the intersection of the circles, x2 + y2 – 6x = 0 and x2 + y2 – 4y = 0, having its centre on the line, 2x – 3y + 12 = 0, also passes through the point :

The diameter of the circle, whose centre lies on
the line x + y = 2 in the first quadrant and which
touches both the lines x = 3 and y = 2, is
_______ .

The number of integral values of k for which
the line, 3x + 4y = k intersects the circle,
x2
+ y2
– 2x – 4y + 4 = 0 at two distinct points is
______.

Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then :

If the circles
x2 + y2 \(-\) 16x \(-\) 20y + 164 = r2  
and  (x \(-\) 4)2 + (y \(-\) 7)2 = 36
intersect at two distinct points, then :

If a tangent to the circle x2 + y2 = 1 intersects
the coordinate axes at distinct points P and Q,
then the locus of the mid-point of PQ is :

The common tangent to the circles x 2 + y2 = 4 and
x2 + y2 + 6x + 8y – 24 = 0 also passes through the
point :

A rectangle is inscribed in a circle with a diameter
lying along the line 3y = x + 7. If the two adjacent
vertices of the rectangle are (–8, 5) and (6, 5), then
the area of the rectangle (in sq. units) is :

The sum of the squares of the lengths of the chords
intercepted on the circle, x2 + y2 = 16, by the lines,
x + y = n, n \(\in\) N, where N is the set of all natural
numbers, is :

The tangent and the normal lines at the point
( \(\sqrt 3\), 1) to the circle x2
+ y2 = 4 and the x-axis form a triangle. The area of this triangle (in
square units) is :

If a variable line, 3x + 4y – \(\lambda\) = 0 is such that the two circles x2 + y2 – 2x – 2y + 1 = 0 and x2 + y2 – 18x – 2y + 78 = 0 are on its opposite sides, then the set of all values of \(\lambda\) is the interval :

Let C1 and C2 be the centres of the circles x2 + y2 – 2x – 2y – 2 = 0 and x2 + y2 – 6x – 6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC1QC2 is :

If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then the
locus of the foot of perpendicular from O on AB is :

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90o, then
the length (in cm) of their common chord is :

A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the
point :

Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is :

A square is inscribed in the circle x2 + y2
– 6x + 8y – 103 = 0 with its sides parallel to the coordinate axes.
Then the distance of the vertex of this square which is nearest to the origin is :

The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is :

If a circle C passing through the point (4, 0) touches the circle x2 + y2 + 4x – 6y = 12 externally at the point (1, – 1), then the radius of C is :

If the area of an equilateral triangle inscribed in the circle x2 + y2
+ 10x + 12y + c = 0 is \(27\sqrt 3\) sq units then c is equal to :

The line x = y touches a circle at the point (1,1). If the circle also passes through the point (1, – 3), then its
radius is :

If the circles x2
+ y2
+ 5Kx + 2y + K = 0 and 2(x2
+ y2) + 2Kx + 3y –1 = 0, (K\(\in\)R), intersect at the points
P and Q, then the line 4x + 5y – K = 0 passes through P and Q, for :

The locus of the centres of the circles, which touch the circle, x2
+ y2
= 1 externally, also touch the y-axis and
lie in the first quadrant, is :

If a circle C, whose radius is 3, touches externally the circle,
\({x^2} + {y^2} + 2x - 4y - 4 = 0\) at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :

A 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 :

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 center is (3, \(-\)2). The radius of C2 is :

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 :