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Two tangents are drawn from a point \( P \) to the circle \( x^{2}+y^{2}-2x-4y+4=0 \), such that the angle between these tangent is \( \tan^{-1}\!\left(\dfrac{12}{5}\right) \), where \( \tan^{-1}\!\left(\dfrac{12}{5}\right)\in(0,\pi) \). If the centre of the circle is denotes by \( C \) and these tangents touch the circle at points \( A \) and \( B \), then the ratio of the areas of \( \triangle PAB \) and \( \triangle CAB \) 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 \( r_{1} \) and \( r_{2} \) be the radii of the largest and smallest circles, respectively, which pass through the point \( (-4,1) \) and have their centres on the circumference of the circle \( x^{2}+y^{2}+2x+4y-4=0 \). If \( \dfrac{r_{1}}{r_{2}}=a+b\sqrt{2} \), then \( a+b \) is equal to:

The minimum distance between any two points \( P_{1} \) and \( P_{2} \) while considering point \( P_{1} \) on one circle and point \( P_{2} \) on the other circle for the given circles' equations \( x^{2} + y^{2} - 10x - 10y + 41 = 0 \) and \( x^{2} + y^{2} - 24x - 10y + 160 = 0 \) is ______________.

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

Let \( \mathbb{Z} \) be the set of all integers, \( A = \{(x,y) \in \mathbb{Z} \times \mathbb{Z} : (x-2)^{2} + y^{2} \leq 4 \} \), \( B = \{(x,y) \in \mathbb{Z} \times \mathbb{Z} : x^{2} + y^{2} \leq 4 \} \) and \( C = \{(x,y) \in \mathbb{Z} \times \mathbb{Z} : (x-2)^{2} + (y-2)^{2} \leq 4 \} \). If the total number of relations from \( A \cap B \) to \( A \cap C \) is \( 2^{p} \), then the value of \( p \) is :

Let \( B \) be the centre of the circle \( x^{2}+y^{2}-2x+4y+1=0 \). Let the tangents at two points \( P \) and \( Q \) on the circle intersect at the point \( A(3,1) \). Then \( 8\left(\dfrac{\text{area } \triangle APQ}{\text{area } \triangle BPQ}\right) \) is equal to ______.

Choose the correct statement about two circles whose equations are given below:
\( x^2 + y^2 - 10x - 10y + 41 = 0 \)
\( x^2 + y^2 - 22x - 10y + 137 = 0 \)

Let \( S_{1} : x^{2} + y^{2} = 9 \) and \( S_{2} : (x - 2)^{2} + y^{2} = 1 \). Then the locus of center of a variable circle \( S \) which touches \( S_{1} \) internally and \( S_{2} \) externally always passes through the points:

Let \( P \) and \( Q \) be two distinct points on a circle which has centre at \( C(2,3) \) and which passes through origin \( O \). If \( OC \) is perpendicular to both the line segments \( CP \) and \( CQ \), then the set \( \{P, Q\} \) is equal to:

Let \( A = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid 2x^2 + 2y^2 - 2x - 2y = 1\} \), \( B = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid 4x^2 + 4y^2 - 16y + 7 = 0\} \) and \( C = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid x^2 + y^2 - 4x - 2y + 5 \le r^2\} \}\).
Then the minimum value of \( |r| \) such that \( A \cup B \subseteq C \) is equal to:

The equation of a circle is \( \operatorname{Re}(z^{2}) + 2\,\operatorname({Im}(z))^{2} + 2\,\operatorname{Re}(z) = 0 \), where \( z = x + iy \). A line which passes through the centre of the given circle and the vertex of the parabola, \( x^{2} - 6x - y + 13 = 0 \), has \( y \) intercept equal to _______.

Consider a circle \( C \) which touches the y-axis at \( (0, 6) \) and cuts off an intercept \( 6\sqrt{5} \) on the x-axis. Then the radius of the circle \( C \) is equal to:

For the four circles \( M \), \( N \), \( O \) and \( P \), following four equations are given:
Circle \( M \): \( x^2 + y^2 = 1 \)
Circle \( N \): \( x^2 + y^2 - 2x = 0 \)
Circle \( O \): \( x^2 + y^2 - 2x - 2y + 1 = 0 \)
Circle \( P \): \( x^2 + y^2 - 2y = 0 \)
If the centre of circle \( M \) is joined with centre of the circle \( N \), further centre of circle \( N \) is joined with centre of the circle \( O \), centre of circle \( O \) is joined with the centre of circle \( P \) and lastly, centre of circle \( P \) is joined with centre of circle \( M \), then these lines form the sides of a:

Let the circle \( S: 36x^2 + 36y^2 - 108x + 120y + c = 0 \) 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 the tangent to the circle \( x^{2} + y^{2} = 25 \) 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 \( OPQ \), then \( r^{2} \) is equal to:

A circle \( C \) touches the line \( x = 2y \) at the point \( (2, 1) \) and intersects the circle \( C_1 : x^2 + y^2 + 2y - 5 = 0 \) at two points \( P \) and \( Q \) such that \( PQ \) is a diameter of \( C_1 \). Then the diameter of \( C \) is:

Let the equation \( x^2 + y^2 + p x + (1 - p) y + 5 = 0 \) represent circles of varying radius \( r \in (0, 5] \). Then the number of elements in the set \( S = \{ q : q = p^2 \text{ and } q \text{ is an integer} \} \) is ____________.

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 \( d^{2} \) is equal to ________.

Two circles each of radius \( 5 \) units touch each other at the point \( (1, 2) \). If the equation of their common tangents is \( 4x + 3y = 10 \), and \( C_{1} (\alpha, \beta) \) and \( C_{2} (\gamma, \delta) \), \( C_{1} \ne C_{2} \) are their centres, then \( |(\alpha + \beta)(\gamma + \delta)| \) 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 \( 4r^2 \) is equal to __________.

If a line along a chord of the circle \( 4x^{2}+4y^{2}+120x+675=0 \), passes through the point \( (-30,0) \), and is tangent to the parabola \( y^{2}=30x \), then the length of this chord is:

Choose the incorrect statement about the two circles whose equations are given below: \( x^{2} + y^{2} - 10x - 10y + 41 = 0 \) and \( x^{2} + y^{2} - 16x - 10y + 80 = 0 \)

If the variable line \( 3x + 4y = \alpha \) lies between the two circles \( (x - 1)^2 + (y - 1)^2 = 1 \) and \( (x - 9)^2 + (y - 1)^2 = 4 \), without intercepting a chord on either circle, then the sum of all the integral values of \( \alpha \) is __________ .

Two tangents are drawn from the point \( P(-1,1) \) to the circle \( x^2 + y^2 - 2x - 6y + 6 = 0 \). If these tangents touch the circle at points \( A \) and \( B \), and if \( D \) is a point on the circle such that length of the segments \( AB \) and \( AD \) are equal, then the area of triangle \( ABD \) is equal to:

A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point \( (2,-3) \) from the line \( 3x+4y=5 \), is given by: