The equation of the locus of a point, which is at a distance of 5 units from a fixed point $(1,4)$ and also from a fixed line $2 x+3 y-1=0$ is

If the equation of the circumcircle of the triangle formed by the lines $L_1 \equiv x+y=0$,

$L_2 \equiv 2 x+y-1=0, L_3 \equiv x-3 y+2=0$ is $\lambda_1 L_1 L_2+\lambda_2 L_2 L_3+\lambda_3 L_3 L_1=0$, then $\frac{7 \lambda_1}{\lambda_2}+\frac{\lambda_3}{\lambda_1}=$

A circle $C$ touches $X$-axis and makes an intercept of length 2 units on $Y$-axis. If the centre of this circle lies on the line $y=x+1$, then a circle passing through the centre of the circle $C$ is

If $m_1, m_2$ are the slopes of the tangents drawn through the point $(-1,-2)$ to the circle $(x-3)^2+(y-4)^2=4$, then $\sqrt{3}\left|m_1-m_2\right|=$