Equation of a circle

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 :

Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1, 1). If the line AP intersects the line BC at the point Q(k$_1$, k$_2$), then k$_1$ + k$_2$ is equal to :

Let the mirror image of a circle $c_1:\ x^2+y^2-2x-6y+\alpha=0$ in line $y=x+1$ be $c_2:\ 5x^2+5y^2+10gx+10fy+38=0$. If r is the radius of circle $c_2$, then $\alpha+6r^2$ is equal to _______.

Let $S=\left\{(x,y)\in\mathbb N\times\mathbb N:9(x-3)^2+16(y-4)^2\le144\right\}$ and $T=\left\{(x,y)\in\mathbb R\times\mathbb R:(x-7)^2+(y-4)^2\le36\right\}$. Then $n(S\cap T)$ is equal to _______.

Consider a triangle $\Delta$ whose two sides lie on the $x$-axis and the line $x + y + 1 = 0$. If the orthocenter of $\Delta$ is $(1, 1)$, then the equation of the circle passing through the vertices of the triangle $\Delta$ is

For any complex number $w = c + id$, let $\arg(w) \in (-\pi, \pi]$, where $i = \sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z = x + iy$ satisfying $\arg\left(\dfrac{z + \alpha}{z + \beta}\right) = \dfrac\pi4$, the ordered pair $(x, y)$ lies on the circle
$$x^2 + y^2 + 5x - 3y + 4 = 0$$
Then which of the following statements is (are) TRUE?

The radius of the circle $C$ is ____.

The value of $\alpha$ is ____.

Consider $M$ with $r = \dfrac{1025}{513}$. Let $k$ be the number of all those circles $C_n$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then

Consider $M$ with $r = \dfrac{(2^{199} - 1)\sqrt2}{2^{198}}$. The number of all those circles $D_n$ that are inside $M$ is

If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord of another circle 'C', whose center is at (2, 1), then its radius is _______.

Let $O$ be the centre of the circle $x^2 + y^2 = r^2$, where $r > \dfrac{\sqrt5}{2}$. Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2x + 4y = 5$. If the centre of the circumcircle of the triangle $OPQ$ lies on the line $x + 2y = 4$, then the value of $r$ is ______

A line $y = mx + 1$ intersects the circle $(x-3)^2 + (y+2)^2 = 25$ at the points $P$ and $Q$. If the midpoint of the line segment $PQ$ has $x$-coordinate $-\dfrac35$, then which one of the following options is correct?

Which of the following is the only CORRECT combination?

Let $T$ be the line passing through the points $P(-2, 7)$ and $Q(2, -5)$. Let $F_1$ be the set of all pairs of circles $(S_1, S_2)$ such that $T$ is tangent to $S_1$ at $P$ and tangent to $S_2$ at $Q$, and also such that $S_1$ and $S_2$ touch each other at a point, say, $M$. Let $E_1$ be the set representing the locus of $M$ as the pair $(S_1, S_2)$ varies in $F_1$. Let the set of all straight line segments joining a pair of distinct points of $E_1$ and passing through the point $R(1, 1)$ be $F_2$. Let $E_2$ be the set of the mid-points of the line segments in the set $F_2$. Then, which of the following statement(s) is (are) TRUE?