The gradient of the line joining the points on the curve $y = x^2 + 2x$ whose abscissae are $1$ and $3$,is

If the coordinates of the points $A$ and $B$ are $(3, 3)$ and $(7, 6)$,then the length of the portion of the line $AB$ intercepted between the axes is

The equation of a given straight line is $\frac{x-x_1}{\cos \theta}=\frac{y-y_1}{\sin \theta}=\gamma$. If the equation of the line perpendicular to the given line and passing through $(\alpha, \beta)$ is $\frac{x}{a}+\frac{y}{b}=1$,then $\frac{b}{a}$ is equal to

If the slope of a line passing through the point $A(3, 2)$ is $3/4$,then the points on the line which are $5$ units away from $A$ are:

The line $MN$ whose equation is $x-y-2=0$ cuts the $X$-axis at $M$ and the coordinates of $N$ are $(4,2)$. The line $MN$ is rotated about $M$ through $45^{\circ}$ in the anticlockwise direction. What is the equation of the line $MN$ in the new position?

If two vertical poles of heights $20\,m$ and $80\,m$ stand apart on a horizontal plane,then the height (in $m$) of the point of intersection of the lines joining the top of each pole to the foot of the other is: