Circles

If one of the diameters of the circle $x^2+y^2-2\sqrt2x-6\sqrt2y+14=0$ is a chord of the circle $\left(x-2\sqrt2\right)^2+\left(y-2\sqrt2\right)^2=r^2$, then the value of $r^2$ is equal to ______.

Let the tangent to the circle $C_1:x^2+y^2=2$ at the point $M(-1,1)$ intersect the circle $C_2:(x-3)^2+(y-2)^2=5$, at two distinct points A and B. If the tangents to $C_2$ at the points A and B intersect at N, then the area of the triangle ANB is equal to :

Let a triangle ABC be inscribed in the circle $x^2-\sqrt2(x+y)+y^2=0$ such that $\angle BAC=\dfrac\pi2$. If the length of side AB is $\sqrt2$, then the area of the $\Delta ABC$ is equal to :

Let $\dfrac{dy}{dx}=\dfrac{ax-by+a}{bx+cy+a}$, $a,b,c\in\mathbf R$, represents a circle with center $(\alpha,\beta)$. Then, $\alpha+2\beta$ is equal to :

Consider three circles :
$$C_1:x^2+y^2=r^2$$
$$C_2:(x-1)^2+(y-1)^2=r^2$$
$$C_3:(x-2)^2+(y-1)^2=r^2$$
If a line $L:y=mx+c$ be a common tangent to $C_1$, $C_2$ and $C_3$ such that $C_1$ and $C_3$ lie on one side of line L while $C_2$ lies on other side, then the value of $20\left(r^2+c\right)$ is equal to :

For $n\in\mathbf N$, let $S_n=\left\{z\in\mathbf C:|z-3+2i|=\dfrac n4\right\}$ and $T_n=\left\{z\in\mathbf C:|z-2+3i|=\dfrac1n\right\}$. Then the number of elements in the set $\{n\in\mathbf N:S_n\cap T_n=\phi\}$ is :

Let the locus of the centre $(\alpha,\beta)$, $\beta>0$, of the circle which touches the circle $x^2+(y-1)^2=1$ externally and also touches the $x$-axis be L. Then the area bounded by L and the line $y=4$ is :

Let the equation of two diameters of a circle $x^2+y^2-2x+2fy+1=0$ be $2px-y=1$ and $2x+py=4p$. Then the slope $m\in(0,\infty)$ of the tangent to the hyperbola $3x^2-y^2=3$ passing through the centre of the circle is equal to ______.

The sum of diameters of the circles that touch (i) the parabola $75x^2=64(5y-3)$ at the point $\left(\dfrac85,\dfrac65\right)$ and (ii) the $y$-axis, is equal to ______.

If the circles $x^2+y^2+6x+8y+16=0$ and $x^2+y^2+2\left(3-\sqrt3\right)x+2\left(4-\sqrt6\right)y=k+6\sqrt3+8\sqrt6$, $k>0$, touch internally at the point $P(\alpha,\beta)$, then $\left(\alpha+\sqrt3\right)^2+\left(\beta+\sqrt6\right)^2$ is equal to ______.

A point P moves so that the sum of squares of its distances from the points (1, 2) and (−2, 1) is 14. Let $f(x,y)=0$ be the locus of P, which intersects the $x$-axis at the points A, B and the $y$-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to :

Let P and Q be any points on the curves $(x-1)^2+(y+1)^2=1$ and $y=x^2$, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval

Let the abscissae of the two points P and Q on a circle be the roots of $x^2-4x-6=0$ and the ordinates of P and Q be the roots of $y^2+2y-7=0$. If PQ is a diameter of the circle $x^2+y^2+2ax+2by+c=0$, then the value of $(a+b-c)$ is ______.

The area of the smaller region enclosed by the curves $y^2=8x+4$ and $x^2+y^2+4\sqrt3x-4=0$ is equal to

Let $P(a,b)$ be a point on the parabola $y^2=8x$ such that the tangent at P passes through the centre of the circle $x^2+y^2-10x-14y+65=0$. Let A be the product of all possible values of $a$ and B be the product of all possible values of $b$. Then the value of $A+B$ is equal to

If the circle $x^2+y^2-2gx+6y-19c=0$, $g,c\in\mathbf R$ passes through the point $(6,1)$ and its centre lies on the line $x-2cy=8$, then the length of intercept made by the circle on $x$-axis is

The equations of the sides AB, BC and CA of a triangle ABC are $2x+y=0$, $x+py=39$ and $x-y=3$ respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true ?

A circle $C_1$ passes through the origin O and has diameter 4 on the positive $x$-axis. The line $y=2x$ gives a chord OA of circle $C_1$. Let $C_2$ be the circle with OA as a diameter. If the tangent to $C_2$ at the point A meets the $x$-axis at P and $y$-axis at Q, then QA : AP is equal to :

For $t\in(0,2\pi)$, if ABC is an equilateral triangle with vertices $A(\sin t,-\cos t)$, $B(\cos t,\sin t)$ and $C(a,b)$ such that its orthocentre lies on a circle with centre $\left(1,\dfrac13\right)$, then $\left(a^2-b^2\right)$ is equal to :

Let C be the centre of the circle $x^2+y^2-x+2y=\dfrac{11}4$ and P be a point on the circle. A line passes through the point C, makes an angle of $\dfrac\pi4$ with the line CP and intersects the circle at the points Q and R. Then the area of the triangle PQR (in unit$^2$) is :