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This mock test of MCQ (Previous Year Questions) - Circle (Competition Level 1) for JEE helps you for every JEE entrance exam.
This contains 24 Multiple Choice Questions for JEE MCQ (Previous Year Questions) - Circle (Competition Level 1) (mcq) to study with solutions a complete question bank.
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QUESTION: 1

The square of the length of tangent from (3, –4) on the circle x^{2} + y^{2} – 4x – 6y + 3 = 0

[AIEEE-2002]

Solution:

S_{1} = 9 + 16 – 12 + 24 + 3 = 40

QUESTION: 2

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

[AIEEE-2003]

Solution:

5 < r + 3

r > 2

|r_{1} - r_{2}I < C_{1} C_{2}

|r- 3| < 5

-5 < r - 3 < 5

-2 < r < 8

So finally 2 < r < 8

QUESTION: 3

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

[AIEEE-2003]

Solution:

πr^{2} = 154, 2x - 3y = 5 x 3

On solving 2x - 3y = 5 & 3x - 4y = 7 we get, x = 1, y = -1

So centre (1, -1)

equation of circle (x - 1)^{2} + (y + 1)^{2} = 7^{2}

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

QUESTION: 4

If a circle passes through the point (a, b) and cuts the circle x^{2} + y^{2} = 4 orthogonally, then the locus of its centre is

-[AIEEE-2004]

Solution:

let (h, k) be centre of circle equation of circle

(x - h)^{2} + (y - k)^{2} = (a - h)^{2} + (b - k)^{2}

x^{2} - 2xh + h^{2} + y^{2} - 2yk + k^{2}

= a^{2} + h^{2} - 2ah + b^{2} + k^{2} - 2bk

x^{2} + y^{2} - 2xh - 2yk + 2ah + 2bk - a^{2} - b^{2} = 0

For orthogonal

2g_{1}g_{2} + 2 f_{1 }f_{2} = c_{1} + c_{2}

2h. 0 2k. 0

= 2ah + 2bk - a^{2} - b^{2} - p^{2} = 0

2ah + 2bk - (a^{2} + b^{2} + p^{2}) = 0

locus of (h, k) is tax + 2by - (a^{2} + b^{2} + p^{2}) = 0

QUESTION: 5

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 -

[AIEEE-2004]

Solution:

Let the equation of the circle is (x - h)^{2} + (y - k)^{2} = k^{2}

(2k)^{2} = (a - b)^{2} + (b - 2)^{2}

(b + q)^{2} = (a - p)^{2} + b^{2} + q^{2} - 2bq

(a - p)^{2} = 4bq

(x - p)^{2} = 4pq

QUESTION: 6

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

[AIEEE-2004]

Solution:

2x + 3y + 1= 0

3x - y - 4 = 0 x 3

11x - 11= 0

x = 1

y = -1

2πr = 10π

r = 5

(x - 1)^{2} + (y + 1)^{2} = 5^{2}

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

QUESTION: 7

If the circles x^{2} + y^{2} + 2ax + cy + a = 0 and x^{2} + y^{2} – 3ax + dy – 1 = 0 intersect in two distinct point P and Q then the lines 5x + by – a = 0 passes through P and Q for -

[AIEEE-2005]

Solution:

PQ is common chord equation of PQ is S_{1} - S_{2} = 0

⇒ 5ax + (c - d) y + a + 1 = 0 also PQ 5x + by - a = 0 both are identical so

By (i) & (iii)

-a^{2} = a + 1

a^{2} + a + 1 = 0

which is always > 0

So option B is correct

QUESTION: 8

A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is -

[AIEEE-2005]

Solution:

Let center be (h, k)

h^{2} + k^{2} - 6k + 9 = k^{2} + 4k + 4

h^{2} = 10k - 5

which is a parabola

QUESTION: 9

If a circle passes through the point (a, b) and cuts the circle x^{2} + y^{2} = p^{2 }orthogonally, then the equation of the locus of its centre is -

[AIEEE-2005]

Solution:

let (h, k) be centre of circle equation of circle

(x - h)^{2} + (y - k)^{2} = (a - h)^{2} + (b - k)^{2}

x^{2} - 2xh + h^{2} + y^{2} - 2yk + k^{2}

= a^{2} + h^{2} - 2ah + b^{2} + k^{2} - 2bk

x^{2} + y^{2} - 2xh - 2yk + 2ah + 2bk - a^{2} - b^{2} = 0

For orthogonal

2 g_{1}g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2} 2h. 0 + 2k. 0

= 2ah + 2bk - a^{2} - b^{2} - p^{2} = 0

2ah + 2bk - (a^{2} + b^{2} + p^{2}) = 0 locus of (h, k) is tax + 2by -(a^{2} + b^{2} + p^{2}) = 0

QUESTION: 10

If the pair of line ax^{2 }+ 2(a + b)xy + by^{2} = 0 lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then

[AIEEE-2005]

Solution:

angle between pair of straight line is given by

ax^{2} + 2(a+b) xy + by^{2} = 0

h = (a+b)

4θ = π ⇒ θ = π/4

(a+b)^{2} = 4(a+b)^{2} - 4ab

3(a+b)^{2} - 4ab = 0

3a^{2} + 3b^{2} + 2ab = 0

QUESTION: 11

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

[AIEEE-2006]

Solution:

QUESTION: 12

The triangle PQR is inscribed in the circle, x^{2}+y^{2 }= 25. If Q and R have co-ordinates (3, 4) & (–4, 3) respectively, then √QPR is equal to

[JEE 2000(Scr.), 1 + 1]

Solution:

x^{2} + y^{2} = 25

∴ angle substended by a chord at centre is double of the angle substanded at point on the opposite side of cirlce.

∠QPR = 4

QUESTION: 13

If the circles, x^{2} + y^{2} + 2x + 2ky + 6 = 0 & x^{2} + y^{2} + 2ky + k = 0 intersect orthogonally, then ‘k’ is

Solution:

x^{2}+y^{2}+2x+2ky+6 = 0

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

cuts orthogonally

⇒ 2.1.0 + 2.k.k = 6 + k

⇒ 2k^{2} - k - 6 = 0

(k - 2) (2k + 3) = 0 k = 2,

QUESTION: 14

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle then 2r equals.

[JEE 2001 (Scr.), 1]

Solution:

In ΔPQR

PQ/2r = tan θ

& In ΔPRR

PR/RS = tan θ

⇒

⇒

QUESTION: 15

Let 2x^{2} + y^{2} – 3xy = 0 be the equation of a pair of tangents drawn from the origin 'O' to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.

Solution:

Pair of lines

2x^{2} - 3xy + y^{2} = 0

In ΔOAC

QUESTION: 16

If the tangent at the point P on the circle x^{2} + y^{2} + 6x +6y = 2 meets the straight line 5x – 2y + 6 = 0 at a point Q on the y-axis, then the length of PQ is

Solution:

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

Q ≡ (0,3)

QUESTION: 17

If a > 2b > 0 then the positive value of m for which i s a common tangent to x^{2} + y^{2} = b^{2} and (x - a)^{2} + y^{2} = b^{2} is

[JEE 2002 (Scr.)]

Solution:

x > 2b > 0

x^{2} + y^{2} = b^{2}, (x - a)^{2} + y^{2} = b^{2}

common tangent

⇒

⇒ b^{2} (m^{2} + 1) = a^{2}m^{2} + b^{2} (1 + m^{2}) - 2abm

m ≠ 0 or (ma)^{2} = (2b)

∴ m > 0 m^{2} (a^{2} - 4b^{2}) = 4b^{2}

⇒

QUESTION: 18

The radius of the circle, having centre at (2, 1), whose one of the chord is a diameter of the circle x_{2} + y_{2} – 2x – 6y + 6 = 0

[JEE 2004(Scr.)]

Solution:

S_{1} ≡ x^{2} + y^{2} - 2x - 6y + 6 = 0

center r_{1} (1, 3), r_{2} = 2

QUESTION: 19

Tangents drawn from the point P(1, 8) to the circle x^{2} + y^{2} – 6x – 4y – 11 = 0 touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is

Solution:

x^{2} + y^{2} - 6x - 4y - 11 = 0

PACD is a cyclic quadrialteral C(3, 2)

(x - 1) (x - 3) + (y -8) (y - 2) = 0

x^{2} - 4x + 3 + y^{2} - 10y + 16 = 0

⇒ x^{2} + y^{2} - 4x - 10y + 19 = 0

QUESTION: 20

The centres of two circles C_{1} and C_{2} each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C_{1} and C_{2} and C be a circle touching circles C_{1} and C_{2 }externally. If a common tangent to C_{1} and C passing through P is also a common tangent to C_{2} and C, then the radius of the circle C is

[JEE 2009]

Solution:

(PM)^{2} = 3^{2} - 1^{2} = 8

PQ^{2} = (r + 1)^{2} - 3^{2}

IN ΔPMQ

PQ^{2} = (PM)^{2} + r^{2}

= 8 + r^{2}

r^{2} + 2r + 1 - 9 = 8 + r^{2}

2r = 16

r = 8

QUESTION: 21

Two parallel chords of a circle of radius 2 are at a distance 3 + 1 apart. If the chords subtend at the center, angles of π/k andk 2π/k , where k > 0, then the value of [k] is

{Note : [k] denotes the largest integer less than or equal to k}

[JEE 2010]

Solution:

∴

⇒ k = 3

QUESTION: 22

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

Solution:

Equation of cirde using family of circle (x - a)^{2} + (y - 2)^{2} + λ x = 0

put (-1, 0) to get λ & check from options

QUESTION: 23

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x – 5y = 20 to the circle x^{2} + y^{2} = 9 is

[JEE 2012]

Solution:

Equation of chord of contact

5ax + 4ay - 20y - 45 = 0

5ax + (4a - 20) y - 45 = 0

equation of chord of mid point hx + ky = h^{2} + k^{2}

put the value of a

QUESTION: 24

A tangent PT is drawn to the circle x^{2} + y^{2} = 4 at the point P (√3, 1) . A straight line L, perpendicular toPT is a tangent to the circle (x – 3)^{2 }+ y^{2} = 1. A possible equation of L is

[JEE 2012]

Solution:

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