Circles

The differential equation of the family of circles passing through the points (0, 2) and (0, −2) is :

Let the tangents at two points A and B on the circle $x^2+y^2-4x+3=0$ meet at origin O (0, 0). Then the area of the triangle OAB is :

Two tangent lines $l_1$ and $l_2$ are drawn from the point (2, 0) to the parabola $2y^2=-x$. If the lines $l_1$ and $l_2$ are also tangent to the circle $(x-5)^2+y^2=r$, then 17r is equal to ______.

Let a line L pass through the point of intersection of the lines $bx+10y-8=0$ and $2x-3y=0$, $b\in\mathbf R-\left\{\frac43\right\}$. If the line L also passes through the point (1, 1) and touches the circle $17(x^2+y^2)=16$, then the eccentricity of the ellipse $\dfrac{x^2}5+\dfrac{y^2}{b^2}=1$ 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 AB be a chord of length 12 of the circle $(x-2)^2+(y+1)^2=\dfrac{169}4$. If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB 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 _______.

Let the area enclosed by the solution curve $y=y(x)$ of the differential equation $\dfrac{dy}{dx}+\dfrac{x-2}{y-b}=2$, b > 0, y(5) = 0 is 13π. Let $y=y(x)$ intersect the y-axis at the points P and Q. If the tangents to $y=y(x)$ at P and Q meet at T, then the area of ΔPTQ is

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?

Let the point $B$ be the reflection of the point $A(2, 3)$ with respect to the line $8x - 6y - 23 = 0$. Let $\Gamma_A$ and $\Gamma_B$ be circles of radii 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_A$ and $\Gamma_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $AC$ is ______

Which of the following is the only CORRECT combination?