The equations of the circles, which touch both the axes and the line $4x+3y=12$ and have centers in the first quadrant, are

Let $S$ be the set of all points $\left(\frac{a}{b}, \frac{c}{d}\right)$ on the circle with radius $1$ centered at $(0,0)$,where $a$ and $b$ are relatively prime integers,$c$ and $d$ are relatively prime integers (that is,$\operatorname{HCF}(a, b) = \operatorname{HCF}(c, d) = 1$),and the integers $b$ and $d$ are even. Then,the set $S$ is:

The equation of the pair of straight lines parallel to the $x$-axis and touching the circle $x^2+y^2-6x-4y-12=0$ is

Which of the following statements are true and which are false? In each case,give a valid reason for your answer.
$p:$ Each radius of a circle is a chord of the circle.

Let the circles $C_1: (x-\alpha)^2 + (y-\beta)^2 = r_1^2$ and $C_2: (x-8)^2 + (y-\frac{15}{2})^2 = r_2^2$ touch each other externally at the point $(6,6)$. If the point $(6,6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2:1$,then $(\alpha+\beta) + 4(r_1^2 + r_2^2)$ equals

The number of possible common tangents that can be drawn to the circles $x^2+y^2+4x-6y-3=0$ and $x^2+y^2+4x-2y+1=0$ is