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This mock test of Test: Circle- 4 for JEE helps you for every JEE entrance exam.
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QUESTION: 1

The triangle PQR is inscribed in the circle x^{2} + y^{2} = 25. If Q and R have coordinates (3, 4) and (–4, 3) respectively, then ∠QPR may be equal to

Solution:

(i.e. angle subtended at the centre of a circle is double the angle subtended in the alternate segment).

Hence (c) is the correct answer.

QUESTION: 2

If the circles x^{2} + y^{2} + 2ax + b = 0 and x^{2} + y^{2} + 2cx + b = 0 touch each other, then

Solution:

Since the circles touch each other, equation of the common tangent is S_{1} – S_{2} = 0 ⇒ x = 0.

Putting x = 0 in the equations of the circles, we get y^{2}+ b = 0.

This equation should have equal roots ⇒ b = 0.

Hence (c) is the correct answer.

QUESTION: 3

Two circles with radii ‘r_{1}’ and ‘r_{2}’, r_{1} > r_{2} __>__ 2, touch each other externally. If ‘θ’ be the angle between the direct common tangents, then

Solution:

Hence (B) is the correct answer.

QUESTION: 4

The common chord of x^{2} + y^{2} – 4x – 4y = 0 and x^{2} + y^{2} = 16 subtends at the origin an angle equal to

Solution:

The equation of the common chord of the circles x^{2}+y^{2}-4x-4y = 0 and x^{2}+y^{2} = 16 is x + y = 4 which meets the circle x2+y2 = 16 at points A(4,0) and B(0,4). Obviously OA ⊥ OB. Hence the common chord AB makes a right angle at the centre of the circle x^{2}+y^{2} = 16.

Hence (d) is the correct answer.

QUESTION: 5

A circle C and the circle x^{2} + y^{2} = 1 are orthogonal and have radical axis parallel to y-axis, then C can be

Solution:

(A) The radical axis of the circle x^{2} + y^{2} = 1 and the circle given in is x = -2, which is | | to y-axis

(B) is y = 2, which is | | to x-axis

(C) is x = 0, which is y-axis

The radical axis with circle in (A) | | to y-axis and the given circle intersect circle given in A orthogonally.

QUESTION: 6

The tangents drawn from the origin to the circle x^{2} + y^{2} + 2gx + 2fy + f^{2} = 0 are perpendicular if

Solution:

Radius of the given circle = |g|. For perpendicular tangents origin should lie on the director circle, i.e. distance of origin from the centre of the given circle

QUESTION: 7

If one of the diameters of the circle x^{2} + y^{2} - 2x - 6y + 6 = 0 is a chord to the circle with centre (2, 1), then the radius of the circle is

Solution:

Centre is (1, 3) and radius = 2

If r = radius of second circle then r^{2} = 2^{2} + (3 - 1)^{2} + (2 - 1)^{2}

⇒ r = 3.

Hence (C) is the correct answer.

QUESTION: 8

Let A_{0} A_{1}A_{2}A_{3} A_{4}A_{5} be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments A_{0}A_{1}, A_{0} A_{2} and A_{0} A_{4} is

Solution:

Hence (C) is the correct answer.

QUESTION: 9

If the point (k + 1, k) lies inside the region bound by the curve and the y-axis, then k belong to the interval

Solution:

Since (k + 1, k) lies inside the region bounded by x

⇒ (k + 1)^{2} + k^{2} - 25 < 0

and k + 1 > 0

⇒ 2k^{2} + 2k - 24 < 0 and k > -1

⇒ -4 < k < 3 and k > -1

⇒ -1 < k < 3

Hence (A) is the correct answer.

QUESTION: 10

A line meets the coordinate axes in A and B, and a circle is circumscribing triangle AOB where O is the origin. If m, n are the distances of the tangents to this circle at the origin from the points A and B respectively, then the diameter of the circle is

Solution:

Since AB is the diameter of the circle.

From figure,

Also equation of the circle is

equation of tangent to this circle at (0, 0) is ax + by = 0 ……(1)

∴ m = length of perpendicular from

and n = length of perpendicular from

Hence (D) is the correct answer.

QUESTION: 11

The locus of the centre of a circle which passes through the origin and cuts off a length 2b from the line x = c, is

Solution:

Let the centre be C (h, k). Since the circle passes through the origin,

Let x = c meets the circle in A and B.

From figure, CB^{2} = CD^{2} + BD^{2}

⇒ OC^{2} = CD^{2} + b^{2} (∴CB=OC and BD=b)

⇒ h^{2} + k^{2} = (h - c)^{2} + b^{2}

⇒ k^{2} + 2ch - c^{2} - b^{2} = 0

⇒ locus is y^{2} + 2cx = b^{2} + c^{2}.

Hence (C) is the correct answer.

QUESTION: 12

If the curves ax^{2} + 4xy + 2y^{2} + x + y + 5 = 0 and ax^{2}+ 6xy + 5y^{2}+ 2 x + 3y + 8 = 0 intersect at four concyclic points then the value of a is

Solution:

Any second degree curve passing through the intersection of the given curves is ax^{2} + 4xy + 2y^{2} + x + y + 5 + λ (ax^{2} + 6xy + 5y^{2} +2x + 3y + 8) = 0

If it is a circle, then coefficient of x^{2} = coefficient of y^{2} and coefficient of xy = 0 a(1+ λ) = 2 + 5λ and 4 + 6λ = 0

Hence (B) is the correct answer.

QUESTION: 13

Equation of chord AB of circle x^{2} + y^{2} = 2 passing through P(2, 2) such that PB/PA = 3, is given by

Solution:

Any line passing through (2, 2) will be of the form

When this line cuts the circle x^{2}+y^{2}=2 , (rcosθ+2)^{2} +(rsinθ+2)^{2} = 2

⇒ r^{2} + 4(sinθ + cosθ)r+6 = 0

now if r_{1} = α, r_{2} = 3α then 4α

= - 4(sinθ + cosθ), 3α^{2} = 6 ⇒ sin2θ = 1⇒ θ = π/4 .

So required chord will be y – 2 = 1 (x –2) ⇒ y = x.

Alternative solution

PA.PN = PT^{2} = 2^{2} + 2^{2} - 2 = 6

PB/PA = 3

From (1) and (2), we have PA = √2, PB = 3 √2

⇒ AB = 2 √2

Now diameter of the circle is 2 √2(as radius is √2)

Hence line passes through the centre ⇒ y = x .

Hence (B) is the correct answer.

QUESTION: 14

If P(2, 8) is an interior point of a circle x^{2} + y^{2} –2x + 4y – p = 0 which neither touches nor intersects the axes, then set for p is

Solution:

For internal point p(2, 8), 4 + 64 – 4 + 32 – p < 0 ⇒ p > 96

and x intercept = therefore 1 + p < 0

⇒ p < -1 and y intercept = ⇒ p < -4

Hence (D) is the correct answer.

QUESTION: 15

Radii of the smallest and the largest circle passing through a point lying on the sides of a rectangle with vertices (± 2, ± 1) and touching the circle x^{2} + y_{2} = 9, are r_{1} and r_{2} respectively. Let d = |r_{1} – r_{2}| then minimum value of d is

Solution:

⇒ |r_{2} - r_{2}| is minimum when x is minimum.

|r_{2} - r_{2}| = 1

QUESTION: 16

If the point (2cosθ, 2sinθ) does not lie in the angle between the lines x + y = 2 and x -y = 2 in which the origin lies, then number of solutions of the equation √2 + cosθ + sinθ = 0 is

Solution:

The point (2 cosθ, 2 sinθ) lies on the circle x^{2} + y2 = 4. From the figure, it is obvious that

Hence no solution

QUESTION: 17

The radical axis of the two distinct circles x^{2} + y^{2} + 2gx + 2fy + c = 0 and 2x^{2} + 2y^{2} + 4x + y + 2c = 0 touches the circle x^{2} + y^{2} - 4x - 4y + 4 = 0. Then the centre of the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 can be

Solution:

The radical axis of given circles is

This line is tangent to the circle x^{2} + y^{2} – 4x – 4y + 4 = 0

⇒ either g = 1 or f = 1/4

QUESTION: 18

Equation of a circle that cuts the circle x^{2} + y^{2} + 2gx + 2fy + c = 0, lines x = – g and y = –f orthogonally, is;

Solution:

Clearly (-g, -f) will be the centre of required circle,

Let the circle be (x + g)^{2} + (y + f)^{2} = d

⇒ x^{2} + y^{2} + 2gx + 2fy + g^{2} + f^{2} –d = 0

It has to cut the given circle orthogonally,

⇒ 2g^{2} + 2f^{2} = c + g^{2} + f^{2} –d

⇒ d = c –g^{2} –f^{2}

QUESTION: 19

The range of values of α for which the line 2y = gx + a is a normal to the circle x^{2} + y^{2} + 2gx +2g y - 2 = 0 for all values of g is

Solution:

The line 2y = gx + β should pass through (–g, –g) so,

–2g = –g^{2} + α ⇒ α = g^{2} – 2g = (g – 1)^{2} – 1 __>__ –1.

QUESTION: 20

The point of intersection of the tangents of the circle x^{2} + y^{2} = 10, drawn at end points of the chord x + y = 2 is

Solution:

Let (h, k) be point of intersection of the tangents Then equation of chord of contact is xh + yk = 10

Compare this with x +y = 2

QUESTION: 21

The line x + y = 5 intersects the circle x^{2} +y^{2} - 6x - 8y + 21 = 0 at points A and B, then the locus of the point C such that AC is perpendicular to BC is

Solution:

Since AB is fixed and AC is perpendicular to BC. So, locus of C is a circle whose diameter is AB. So, family of circles passing through AB is

x^{2} + y^{2} - 6x - 8y + 21+λ(x + y -5) = 0

⇒x^{2} + y^{2} - (6 - λ)x - (8 - λ)y + 21- 5λ = 0

Since AB is diameter so centre must lie on AB

∴ Locus is x^{2} + y^{2} - 4x - 6y + 11 = 0.

QUESTION: 22

If a circle of radius 3 units is touching the lines in the first quadrant then length of chord of contact to this circle is

Solution:

Given equation of lines

∠APO = 75°

Length of chord of contact AB

= 6(sin45°cos30° + sin30°cos45°)

QUESTION: 23

(x – 1)(y – 2) = 5 and (x – 1)^{2} + (y + 2)^{2} = r^{2} intersect at four points A, B, C, D and if centroid of ΔABC lies on line y = 3x - 4, then locus of D is

Solution:

If (x_{i}, y_{i}) is the point of intersection of given curves

⇒ y_{4} = 3x_{4} ⇒ locus of D is y = 3x.

QUESTION: 24

Tangents PA and PB are drawn to circle (x - 5)^{2} + (y - 7)^{2} = 1 from point P lying on Locus of circumcentre of triangle PAB is

Solution:

now if centre of circle is(5, 7) ≡ C say then PACB is a cyclic quadrilateral, so circumcentre of triangle PAB is midpoint of PC i.e.

⇒ 4x^{2} - 4xy - 6xy + 10y - 10 = 0

QUESTION: 25

The point on the straight line x + y = 2, which is nearest to the circle x^{2} + y^{2} –10x + 2y + 22 = 0

Solution:

Shortest distance between 2 curves occurs along the common normal. Let required point be (t, 2 -t). Normal at this will be y –(2 –t) = (x –t), it should passes through centre of circle so –1 –2 + t = 5 –t ⇒ t = 4, so point can be (4, -2).

QUESTION: 26

Formula used to measure circumference of circle is

Solution:

QUESTION: 27

In formula 2πr, 'r' is considered as

Solution:

QUESTION: 28

If circumference of circle is 64π then area of circle (in terms of π) is

Solution:

QUESTION: 29

Formula used to measure area of circle is

Solution:

QUESTION: 30

If circumference of circle is 82π then value of 'r' is

Solution:

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