Circle · Mathematics · TS EAMCET

$P$ and $Q$ are the points of trisection of the line segment joining the points $(3,-7)$ and $(-5,3)$. If $P Q$ subtends right angle at a variable point $R$, then the locus of $R$ is

If $A(1,2), B(2,1)$ are two vertices of an acute angled triangle and $S(0,0)$ is its circumcenter, then the angle subtended by $A B$ at the third vertex is

A circle passing through the points $(1,1)$ and $(2,0)$ touches the line $3 x-y-1=0$. If the equation of this circle is $x^{2}+y^{2}+2 g x+2 f y+c=0$, then a possible value of $g$ is

A circle passes through the points $(2,0)$ and $(1,2)$. If the power of the point $(0,2)$ with respect to this circle is 4 , then the radius of the circle is

$x-2 y-6=0$ is a normal to the circle $x^{2}+y^{2}+2 g x+2 f y-8=0$. If the line $y=2$ touches this circle, then the radius of the circle can be

The line $x+y+1=0$ intersects the circle $x^{2}+y^{2}-4 x+2 y-4=0$ at the points $A$ and $B$. If $M(a, b)$ is the mid-point of $A B$, then $a-b=$

A circle $S$ passes through the points of intersection of the circles $x^{2}+y^{2}-2 x-3=0$ and $x^{2}+y^{2}-2 y=0$. If $x+y+1=0$ is a tangent to the circle $S$, then equation of $S$ is

If the common chord of the circles $x^{2}+y^{2}-2 x+2 y+1=0$ and $x^{2}+y^{2}-2 x-2 y-2=0$ is the diameter of a circle $S$, then the center of the circles is

A rhombus is inscribed in the region common to the two circles $x^{2}+y^{2}-4 x-12=0$ and $x^{2}+y^{2}+4 x-12=0$. If the line joining the centres of these circles and the common chord of them are the diagonals of this rhombus, then the area (in sq units) of the rhombus is

If $m$ is the slope and $P(8, \beta)$ is the mid-point of a chord of contact of the circle $x^{2}+y^{2}=125$, then the number of values of $\beta$ such that $\beta$ and $m$ are integers is

A rectangle is formed by the lines $x=4, x=-2, y=5, y=-2$ and a circle is drawn through the vertices of this rectangle. The pole of the line $y+2=0$ with respect to this circle is

The equation of a circle which passes through the points of intersection of the circles $2 x^{2}+2 y^{2}-2 x+6 y-3=0, x^{2}+y^{2}+4 x+2 y+1=0$ and whose centre lies on the common chord of these circles is

If the equation of the circle which cuts each of the circles $x^{2}+y^{2}=4, x^{2}+y^{2}-6 x-8 y+10=0$ and $x^{2}+y^{2}+2 x-4 y-2=0$ at the extremities of a diameter of these circles is $x^{2}+y^{2}+2 g x+2 f y+c=0$, then $g+f+c=$

The equation of the circle passing through the origin and cutting the circles $x^{2}+y^{2}+6 x-15=0$ and $x^{2}+y^{2}-8 y-10=0$ orthogonally is

$(1, k)$ is a point on the circle passing through the points $(-1,1),(0,-1)$ and $(1,0)$. If $k \neq 0$, then $k=$

If the tangents $x+y+k=0$ and $x+a y+b=0$ drawn to the circle $S=x^2+y^2+2 x-2 y+1=0$ are perpendicular to each other and $k, b$ are both greater than 1 , then $b-k=$

If $(h, k)$ is the internal centre of similitude of the circles $x^2+y^2+2 x-6 y+1=0$ and $x^2+y^2-4 x+2 y+4=0$, then $4 h=$

The slope of a common tangent to the circles $x^2+y^2-4 x-8 y+16=0$ and $x^2+y^2-6 x-16 y+64=0$ is

$x^2+y^2+2 x-6 y-6=0$ and $x^2+y^2-6 x-2 y+k=0$ are two intersecting circles and $k$ is not an integer. If $\theta$ is the angle between the two circles and $\cos \theta=\frac{-5}{24}$, then $k=$

If $(p, q)$ is the centre of the circle which cuts the three circles $x^2+y^2-2 x-4 y+4=0, x^2+y^2+2 x-4 y+1=0$ and $x^2+y^2-4 x-2 y-11=0$ orthogonally, then $p+q=$

If $P\left(\frac{\pi}{4}\right), Q\left(\frac{\pi}{3}\right)$ are two points on the circle $x^2+y^2-2 x-2 y-1=0$, then the slope of the tangent to this circle which is parallel to the chord $P Q$ is

The power of a point $(2,0)$ with respect to a circle $S$ is -4 and the length of the tangent drawn from the point $(1,1)$ to $S$ is 2 . If the circle $S$ passes through the point $(-1,-1)$, then the radius of the circle $S$ is

The pole of the line $x-5 y-7=0$ with respect to the circle $S \equiv x^2+y^2-2 x+4 y+1=0$ is $P(a, b)$. If $C$ is the centre of the circle $S=0$, then $P C=$

The equation of the pair of transverse common tangents drawn to the circles $x^2+y^2+2 x+2 y+1=0$ and $x^2+y^2-2 x-2 y+1=0$ is

If a circle passing through the point $(1,1)$ cuts the circles $x^2+y^2+4 x-5=0$ and $x^2+y^2-4 y+3=0$ orthogonally, then the centre of that circle is

Length of the common chord of the circles $x^2+y^2-6 x+5=0$ and $x^2+y^2+4 y-5=0$ is

The centroid of a variable $\triangle A B C$ is at the distance of 5 units from the origin. If $A=(2,3)$ and $B=(3,2)$, then the locus of $C$ is

If $(1,1),(-2,2)$ and $(2,-2)$ are 3 points on a circle $S$, then the perpendicular distance from the centre of the circle $S$ to the line $3 x-4 y+1=0$ is

If the line $4 x-3 y+p=0(p+3>0)$ touches the circle $x^2+y^2-4 x+6 y+4=0$ at the point $(h, k)$, then $h-2 k=$

If the inverse point of the point $P(3,3)$ with respect to the circle $x^2+y^2-4 x+4 y+4=0$ is $Q(a, b)$, then $a+5 b=$

If the equation of the transverse common tangent of the circles $x^2+y^2-4 x+6 y+4=0$ and $x^2+y^2+2 x-2 y-2=0$ is $a x+b y+c=0$, then $\frac{a}{c}=$

A circle $S \equiv x^2+y^2+2 g x+2 f y+6=0$ cuts another circle $x^2+y^2-6 x-6 y-6=0$ orthogonally. If the angle between the circles $S=0$ and $x^2+y^2+6 x+6 y+2=0$ is $60^{\circ}$, then the radius of the circle $S=0$ is

If $m_1$ and $m_2$ are the slopes of the direct common tangents drawn to the circles $x^2+y^2-2 x-8 y+8=0$ and $x^2+y^2-8 x+15=0$, then $m_1+m_2=$

If a diameter of the circle $x^2+y^2-4 x+6 y-12=0$ is a chord of a circle $S$ whose centre is at $(-3,2)$, then the radius of $S$ is

If a circle passing through $A(1,1)$ touches the $X$-axis, then the locus of the other end of the diameter through $A$ is

If $C(\alpha, \beta)(a<0)$ is the centre of the circle that touches the $Y$-axis at $(0,3)$ and makes an intercept of length 2 units on positive $X$-axis, then $(\alpha, \beta)=$

The equations of the tangents to the circle $x^2+y^2=4$ drawn from the point $(4,0)$ are

The image of every point lying on the curve $x^2+y^2=1$ in the line $x+y=1$ satisfies the equation

If the inverse of $P(-3,5)$ with respect to a circle is $(1,3)$ then polar of $P$ with respect to that circle is

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