If the point $$(a, 8,-2)$$ divides the line segment joining the points $$(1,4,6)$$ and $$(5,2,10)$$ in the ratio $$m: n$$, then $$\frac{2 m}{n}-\frac{a}{3}=$$

If $$(a, b, c)$$ are the direction ratios of a line joining the points $$(4,3,-5)$$ and $$(-2,1,-8)$$, then the point $$P(a, 3 b, 2 c)$$ lies on the plane

The $$x$$-intercept of a plane $$\pi$$ passing through the point $$(1,1,1)$$ is $$\frac{5}{2}$$ and the perpendicular distance from the origin to the plane $$\pi$$ is $$\frac{5}{7}$$. If the $$y$$-intercept of the plane $$\pi$$ is negative and the $$z$$-intercept is positive, then its $$y$$-intercept is

The equation of the plane passing through $$3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$$ and parallel to the vectors $$2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$$ is