Standard and General Equation of a Circle

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JEE Main 2016

LEVELJEE Advanced

The centres of those circles which touch the circle, $x^{2} + y^{2} - 8 x - 8 y - 4 = 0$ , externally and also touch the x-axis, lie on:

an ellipse which is not a circle

JEE Main 2016

LEVELJEE Main

If one of the diameters of the circle, given by the equation, $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ , is a chord of a circle $S$ , whose centre is at $(- 3, 2)$ , then the radius of $S$ is:

(A)

(B)

(C)

52

(D)

53

JEE Main 2015

LEVELJEE Main

The number of common tangents to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ , is :

Let $C$ be the circle with centre at $(1, 1)$ and radius $= 1$ . If $T$ is the circle centred at $(0, y)$ , passing through origin and touching the circle $C$ externally, then the radius of $T$ is equal to

(A)

\frac{1}{2}

(B)

\frac{1}{4}

(C)

\frac{3}{2}

(D)

\frac{3}{4}

JEE Main 2013

LEVELJEE Main

The circle passing through $(1, - 2)$ and touching the axis of $x$ at $(3, 0)$ also passes through the point

(A)

(- 5, 2)

(B)

(2, - 5)

(C)

(5, - 2)

(D)

(- 2, 5)

JEE Main 2012

LEVELJEE Main

The length of the diameter of the circle which touches the x-axis at the point $(1, 0)$ and passes through the point $(2, 3)$ is:

(A)

\frac{10}{3}

(B)

\frac{3}{5}

(C)

\frac{6}{5}

(D)

\frac{5}{3}

JEE Main 2011

LEVELJEE Main

The two circles $x^{2} + y^{2} = a x$ and $x^{2} + y^{2} = c^{2}$ ( $c > 0$ ) touch each other if

(A)

∣ a ∣ = c

(B)

a = 2 c

(C)

∣ a ∣ = 2 c

(D)

2∣ a ∣ = c

JEE Main 2009

LEVELJEE Main

Three distinct points $A, B$ and $C$ are given in the 2-dimensional coordinates plane such that the ratio of the distance of any one of them from the point $(1, 0)$ to the distance from the point $(- 1, 0)$ is equal to $\frac{1}{3}$ . Then the circumcentre of the triangle $A B C$ is at the point

(A)

(\frac{5}{4}, 0)

(B)

(\frac{5}{2}, 0)

(C)

(\frac{5}{3}, 0)

(D)

(0, 0)

JEE Main 2008

LEVELBoard

The point diametrically opposite to the point $P (1, 0)$ on the circle $x^{2} + y^{2} + 2 x + 4 y - 3 = 0$ is

(A)

(3, - 4)

(B)

(- 3, 4)

(C)

(- 3, - 4)

(D)

(3, 4)

JEE Main 2007

LEVELJEE Main

Consider a family of circles which are passing through the point $(- 1, 1)$ and are tangent to x-axis. If $(h, k)$ are the coordinate of the centre of the circles, then the set of values of $k$ is given by the interval

(A)

- \frac{1}{2} \leq k \leq \frac{1}{2}

(B)

k \leq \frac{1}{2}

(C)

0 \leq k \leq \frac{1}{2}

(D)

k \geq \frac{1}{2}

JEE Main 2006

LEVELJEE Main

If the lines $3 x - 4 y - 7 = 0$ and $2 x - 3 y - 5 = 0$ are two diameters of a circle of area $49 π$ square units, the equation of the circle is

(A)

x^{2} + y^{2} + 2 x - 2 y - 47 = 0

(B)

x^{2} + y^{2} + 2 x - 2 y - 62 = 0

(C)

x^{2} + y^{2} - 2 x + 2 y - 62 = 0

(D)

x^{2} + y^{2} - 2 x + 2 y - 47 = 0

JEE Main 2006

LEVELJEE Main

Let $C$ be the circle with centre $(0, 0)$ and radius 3 units. The equation of the locus of the mid points of the chords of the circle $C$ that subtend an angle of $\frac{2 π}{3}$ at its center is

(A)

x^{2} + y^{2} = \frac{3}{2}

(B)

x^{2} + y^{2} = 1

(C)

x^{2} + y^{2} = \frac{27}{4}

(D)

x^{2} + y^{2} = \frac{9}{4}

JEE Main 2004

LEVELJEE Main

A variable circle passes through the fixed point $A (p, q)$ and touches x-axis. The locus of the other end of the diameter through $A$ is

(A)

(y - q)^{2} = 4 p x

(B)

(x - q)^{2} = 4 p y

(C)

(y - p)^{2} = 4 q x

(D)

(x - p)^{2} = 4 q y

JEE Main 2004

LEVELJEE Main

If the lines $2 x + 3 y + 1 = 0$ and $3 x - y - 4 = 0$ lie along diameter of a circle of circumference $10 π$ , then the equation of the circle is

(A)

x^{2} + y^{2} + 2 x - 2 y - 23 = 0

(B)

x^{2} + y^{2} - 2 x - 2 y - 23 = 0

(C)

x^{2} + y^{2} + 2 x + 2 y - 23 = 0

(D)

x^{2} + y^{2} - 2 x + 2 y - 23 = 0

JEE Main 2004

LEVELJEE Main

Intercept on the line $y = x$ by the circle $x^{2} + y^{2} - 2 x = 0$ is $A B$ . Equation of the circle on $A B$ as a diameter is

(A)

x^{2} + y^{2} + x - y = 0

(B)

x^{2} + y^{2} - x + y = 0

(C)

x^{2} + y^{2} + x + y = 0

(D)

x^{2} + y^{2} - x - y = 0

JEE Main 2003

LEVELJEE Main

If the two circles $(x - 1)^{2} + (y - 3)^{2} = r^{2}$ and $x^{2} + y^{2} - 8 x + 2 y + 8 = 0$ intersect in two distinct point, then

(A)

r > 2

(B)

2 < r < 8

(C)

r < 2

(D)

r = 2

JEE Main 2003

LEVELJEE Main

The lines $2 x - 3 y = 5$ and $3 x - 4 y = 7$ are diameters of a circle having area as 154 sq.units. Then the equation of the circle is

(A)

x^{2} + y^{2} - 2 x + 2 y = 62

(B)

x^{2} + y^{2} + 2 x - 2 y = 62

(C)

x^{2} + y^{2} + 2 x - 2 y = 47

(D)

x^{2} + y^{2} - 2 x + 2 y = 47

JEE Main 2002

LEVELJEE Main

The centres of a set of circles, each of radius 3, lie on the circle $x^{2} + y^{2} = 25$ . The locus of any point in the set is

(A)

4 \leq x^{2} + y^{2} \leq 64

(B)

x^{2} + y^{2} \leq 25

(C)

x^{2} + y^{2} \geq 25

(D)

3 \leq x^{2} + y^{2} \leq 9

JEE Main 2002

LEVELJEE Main

The centre of the circle passing through $(0, 0)$ and $(1, 0)$ and touching the circle $x^{2} + y^{2} = 9$ is

(A)

(\frac{1}{2}, \frac{1}{2})

(B)

(\frac{1}{2}, - 2)

(C)

(\frac{3}{2}, \frac{1}{2})

(D)

(\frac{1}{2}, \frac{3}{2})

JEE Main 2002

LEVELJEE Main

The equation of a circle with origin as a centre and passing through equilateral triangle whose median is of length $3 a$ is

(A)

x^{2} + y^{2} = 9 a^{2}

(B)

x^{2} + y^{2} = 16 a^{2}

(C)

x^{2} + y^{2} = 4 a^{2}

(D)

x^{2} + y^{2} = a^{2}

JEE Advanced 2013

LEVELJEE Main

Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length $27$ on y-axis is (are)

* Multiple Correct Options

(A)

x^{2} + y^{2} - 6 x + 8 y + 9 = 0

(B)

x^{2} + y^{2} - 6 x + 7 y + 9 = 0

(C)

x^{2} + y^{2} - 6 x - 8 y + 9 = 0

(D)

x^{2} + y^{2} - 6 x - 7 y + 9 = 0

JEE Advanced 2011

LEVELJEE Main

The circle passing through the point $(- 1, 0)$ and touching the y-axis at $(0, 2)$ also passes through the point

(A)

(- \frac{3}{2}, 0)

(B)

(- \frac{5}{2}, 2)

(C)

(- \frac{3}{2}, \frac{5}{2})

(D)

(- 4, 0)

JEE Advanced 2008

LEVELJEE Advanced

A straight line through the vertex $P$ of a triangle $P QR$ intersects the side $QR$ at the point $S$ and the circumcircle of the triangle $P QR$ at the point $T$ . If $S$ is not the centre of the circumcircle, then

* Multiple Correct Options

(A)

\frac{1}{P S} + \frac{1}{S T} < \frac{2}{QS \times S R}

(B)

\frac{1}{P S} + \frac{1}{S T} > \frac{2}{QS \times S R}

(C)

\frac{1}{P S} + \frac{1}{S T} < \frac{4}{QR}

(D)

\frac{1}{P S} + \frac{1}{S T} > \frac{4}{QR}

JEE Advanced 2008

LEVELJEE Main

Let $a$ and $b$ be non-zero real numbers. Then, the equation $(a x^{2} + b y^{2} + c) (x^{2} - 5 x y + 6 y^{2}) = 0$ represents

(A)

four straight lines, when

c = 0

and

a, b

are of the same sign.

(B)

two straight lines and a circle, when

a = b

, and

c

is of sign opposite to that of

a

(C)

two straight lines and a hyperbola, when

a

and

b

are of the same sign and

c

is of sign opposite to that of

a

(D)

a circle and an ellipse, when

a

and

b

are of the same sign and

c

is of sign opposite to that of

a

LEVELJEE Advanced

A circle is given by $x^{2} + (y - 1)^{2} = 1$ , another circle $C$ touches it externally and also the x-axis, then the locus of its centre is

(A)

{(x, y) : x^{2} = 4 y} \cup {(x, y) : y \leq 0}

(B)

{(x, y) : x^{2} + (y - 1)^{2} = 4} \cup {(x, y) : y \leq 0}

(C)

{(x, y) : x^{2} = y} \cup {(x, y) : y \leq 0}

(D)

{(x, y) : x^{2} = 4 y} \cup {(0, y) : y \leq 0}

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The centres of those circles which touch the circle, x2+y2−8x−8y−4=0, externally and also touch the x-axis, lie on:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one of the diameters of the circle, given by the equation, x2+y2−4x+6y−12=0, is a chord of a circle S, whose centre is at (−3,2), then the radius of S is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of common tangents to the circles x2+y2−4x−6y−12=0 and x2+y2+6x+18y+26=0, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The circle passing through (1,−2) and touching the axis of x at (3,0) also passes through the point

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The length of the diameter of the circle which touches the x-axis at the point (1,0) and passes through the point (2,3) is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The two circles x2+y2=ax and x2+y2=c2 (c>0) touch each other if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The point diametrically opposite to the point P(1,0) on the circle x2+y2+2x+4y−3=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the lines 3x−4y−7=0 and 2x−3y−5=0 are two diameters of a circle of area 49π square units, the equation of the circle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A variable circle passes through the fixed point A(p,q) and touches x-axis. The locus of the other end of the diameter through A is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the lines 2x+3y+1=0 and 3x−y−4=0 lie along diameter of a circle of circumference 10π, then the equation of the circle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Intercept on the line y=x by the circle x2+y2−2x=0 is AB. Equation of the circle on AB as a diameter is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the two circles (x−1)2+(y−3)2=r2 and x2+y2−8x+2y+8=0 intersect in two distinct point, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The lines 2x−3y=5 and 3x−4y=7 are diameters of a circle having area as 154 sq.units. Then the equation of the circle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The centres of a set of circles, each of radius 3, lie on the circle x2+y2=25. The locus of any point in the set is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The centre of the circle passing through (0,0) and (1,0) and touching the circle x2+y2=9 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of a circle with origin as a centre and passing through equilateral triangle whose median is of length 3a is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length 27​ on y-axis is (are)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The circle passing through the point (−1,0) and touching the y-axis at (0,2) also passes through the point

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a and b be non-zero real numbers. Then, the equation (ax2+by2+c)(x2−5xy+6y2)=0 represents

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A circle is given by x2+(y−1)2=1, another circle C touches it externally and also the x-axis, then the locus of its centre is

The centres of those circles which touch the circle, $x^{2} + y^{2} - 8 x - 8 y - 4 = 0$ , externally and also touch the x-axis, lie on:

If one of the diameters of the circle, given by the equation, $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ , is a chord of a circle $S$ , whose centre is at $(- 3, 2)$ , then the radius of $S$ is:

The number of common tangents to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ , is :

The circle passing through $(1, - 2)$ and touching the axis of $x$ at $(3, 0)$ also passes through the point

The length of the diameter of the circle which touches the x-axis at the point $(1, 0)$ and passes through the point $(2, 3)$ is:

The two circles $x^{2} + y^{2} = a x$ and $x^{2} + y^{2} = c^{2}$ ( $c > 0$ ) touch each other if

The point diametrically opposite to the point $P (1, 0)$ on the circle $x^{2} + y^{2} + 2 x + 4 y - 3 = 0$ is

If the lines $3 x - 4 y - 7 = 0$ and $2 x - 3 y - 5 = 0$ are two diameters of a circle of area $49 π$ square units, the equation of the circle is

A variable circle passes through the fixed point $A (p, q)$ and touches x-axis. The locus of the other end of the diameter through $A$ is

If the lines $2 x + 3 y + 1 = 0$ and $3 x - y - 4 = 0$ lie along diameter of a circle of circumference $10 π$ , then the equation of the circle is

Intercept on the line $y = x$ by the circle $x^{2} + y^{2} - 2 x = 0$ is $A B$ . Equation of the circle on $A B$ as a diameter is

If the two circles $(x - 1)^{2} + (y - 3)^{2} = r^{2}$ and $x^{2} + y^{2} - 8 x + 2 y + 8 = 0$ intersect in two distinct point, then

The lines $2 x - 3 y = 5$ and $3 x - 4 y = 7$ are diameters of a circle having area as 154 sq.units. Then the equation of the circle is

The centres of a set of circles, each of radius 3, lie on the circle $x^{2} + y^{2} = 25$ . The locus of any point in the set is

The centre of the circle passing through $(0, 0)$ and $(1, 0)$ and touching the circle $x^{2} + y^{2} = 9$ is

The equation of a circle with origin as a centre and passing through equilateral triangle whose median is of length $3 a$ is

Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length $27$ on y-axis is (are)

The circle passing through the point $(- 1, 0)$ and touching the y-axis at $(0, 2)$ also passes through the point

Let $a$ and $b$ be non-zero real numbers. Then, the equation $(a x^{2} + b y^{2} + c) (x^{2} - 5 x y + 6 y^{2}) = 0$ represents

A circle is given by $x^{2} + (y - 1)^{2} = 1$ , another circle $C$ touches it externally and also the x-axis, then the locus of its centre is