Let $$A(a, b), B(3,4)$$ and $$C(-6,-8)$$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $$P(2 a+3,7 b+5)$$ from the line $$2 x+3 y-4=0$$ measured parallel to the line $$x-2 y-1=0$$ is

Let $$\alpha, \beta, \gamma, \delta \in \mathbb{Z}$$ and let $$A(\alpha, \beta), B(1,0), C(\gamma, \delta)$$ and $$D(1,2)$$ be the vertices of a parallelogram $$\mathrm{ABCD}$$. If $$A B=\sqrt{10}$$ and the points $$\mathrm{A}$$ and $$\mathrm{C}$$ lie on the line $$3 y=2 x+1$$, then $$2(\alpha+\beta+\gamma+\delta)$$ is equal to

If $$x^2-y^2+2 h x y+2 g x+2 f y+c=0$$ is the locus of a point, which moves such that it is always equidistant from the lines $$x+2 y+7=0$$ and $$2 x-y+8=0$$, then the value of $$g+c+h-f$$ equals

A line passing through the point $$\mathrm{A}(9,0)$$ makes an angle of $$30^{\circ}$$ with the positive direction of $$x$$-axis. If this line is rotated about A through an angle of $$15^{\circ}$$ in the clockwise direction, then its equation in the new position is :