Circle Equation & Properties: JEE Questions & Solutions

Let PQ and MN be two straight lines touching the circle x²+y²-4x-6y-3=0 at the points A and B respectively. Let O be the centre of the…

If P is a point on the circle x² + y² = 4, Q is a point on the straight line 5x + y + 2 = 0 and x - y + 1 = 0 is the perpendicular bisector…

Let the set of all values of r, for which the circles (x + 1)² + (y + 4)² = r² and x² + y² - 4x - 2y - 4 = 0 intersect at two distinct…

Let a circle of radius 4 pass through the origin O, the points A(-√3a,0) and B(0,-√2b), where a and b are real parameters and ab≠0. Then…

Let y = x be the equation of a chord of the circle C₁ (in the closed half-plane x ≥ 0) of diameter 10 passing through the origin. Let C₂ be…

Let the circle x²+y²=4 intersect the x-axis at points A(a,0) and B(b,0). Let P(2cosα,2sinα), 0<α<π/2, and Q(2cosβ,2sinβ) be two points on…

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at…

Let circle C be the image of x² + y² - 2x + 4y - 4 = 0 in the line 2x - 3y + 5 = 0 and A be the point on C such that OA is parallel to the…

Let the equation of the circle, which touches the x-axis at the point (a, 0), a > 0, and cuts off an intercept of length b on the y-axis be…

Let the line x + y = 1 meet the circle x² + y² = 4 at the points A and B. If the line perpendicular to AB and passing through the mid point…

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord of the circle…

The absolute difference between the squares of the radii of the two circles passing through the point (-9, 4) and touching the lines x + y…

If the four distinct points (4, 6), (-1, 5), (0, 0) and (k, 3k) lie on a circle of radius r, then 10k + r² is equal to

Let C₁ be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let C₂ be the circle with center (1, 3) that…

Four distinct points (2k,3k), (1,0), (0,1) and (0,0) lie on a circle for k equal to:

Consider a circle (x - α)² + (y - β)² = 50, where α, β > 0. If the circle touches the line y + x = 0 at point P, whose distance from the…

Equation of two diameters of a circle are 2x - 3y = 5 and 3x - 4y = 7. The line joining the points (-22/7, -4) and (-1/7, 3) intersects the…

The area (in sq. units) of the part of the circle x² + y² = 169 which is below the line 5x - y = 13 is πα/2β - 65/2 + α/βsin⁻¹(12/13),…

If the circles (x + 1)² + (y +2)² = r² and x² + y² - 4x - 4y + 4 = 0 intersect at exactly two distinct points, then:

Consider two circles C₁: x² + y² = 25 and C₂: (x-α)² + y² = 16, where α ∈ (5, 9). Let the angle between the two radii (one to each circle)…

If one of the diameters of the circle x² + y² - 10x + 4y + 13 = 0 is a chord of another circle C, whose center is the point of intersection…

Let the circles C₁: (x - α)² + (y - β)² = r₁² and C₂: (x - 8)² + (y - 15/2)² = r₂² touch each other externally at the point (6,6). If the…

If the image of the point (-4, 5) in the line x + 2y = 2 lies on the circle (x + 4)² + (y - 3)² = r², then r is equal to:

The locus of the midpoints of the chords of the circle C₁: (x - 4)² + (y - 5)² = 4, which subtend an angle θᵢ at the centre of the circle…

The points of intersection of the line ax + by = 0, (a ≠ b) and the circle x² + y² - 2x = 0 are A(α, 0) and B(1, β). The image of the…

Points P(-3, 2), Q(9, 10), and R(α, 4) lie on a circle C with PR as its diameter. The tangents to C at Q and R intersect at point S. If S…

Let the tangents at the points A((4, -11)) and B((8, -5)) on the circle x² + y² - 3x + 10y - 15 = 0 intersect at the point C. Then the…

A circle with center (2, 3) and radius 4 intersects the line x + y = 3 at points P and Q. If the tangents at P and Q intersect at S(α, β),…

Let y = x + 2, 4y = 3x + 6, and 3y = 4x + 1 be three tangent lines to the circle (x - h)² + (y - k)² = r². Then h + k is equal to:

Let a circle C₁ be obtained on rolling the circle x² + y² - 4x - 6y + 11 = 0 upwards 4 units on the tangent T to it at the point (3, 2).…

If the center and radius of the circle | (z - 2)/(z - 3) | = 2 are respectively (α, β) and γ, then 3(α + β + γ) is equal to:

A circle passing through the point P(α,β) in the first quadrant touches the two coordinate axes at the points A and B. The point P is above…

If the tangents at the points P and Q on the circle x² + y² - 2x + y = 5 meet at the point R(9/4, 2), then the area of triangle PQR is:

Consider a circle C₁: x² + y² - 4x - 2y = α - 5. Let its mirror image in the line y = 2x + 1 be another circle C₂: 5x² + 5y² - 10fx - 10gy…

An arc PQ of a circle subtends a right angle at its centre O. The midpoint of the arc PQ is R. If OP=u, OR=v and OQ=α u+β v, then α, β² are…

A line segment AB of length λ moves such that the points A and B remain on the periphery of a circle of radius λ. Then the locus of the…

Let A be the point (1, 2) and B be any point on the curve x² + y² = 16. If the centre of the locus of the point P, which divides the line…

Two circles in the first quadrant of radii r₁ and r₂ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line…

Let the centre of a circle C be (α, β) and its radius r < 8. Let 3x + 4y = 24 and 3x - 4y = 32 be two tangents and 4x + 3y = 1 be a normal…

The number of common tangents to the circles x²+y²-18x-15y+131=0 and x²+y²-6x-6y-7=0 is