What is the equation of the line passing through $(-1, 1)$ and perpendicular to the line $2x + 3y + 4 = 0$?

Reduce the equation $\sqrt{3} x + y - 8 = 0$ into normal form. Find the values of $p$ and $\omega$.

Let the coordinates of the two points $A$ and $B$ be $(1, 2)$ and $(7, 5)$ respectively. The line $AB$ is rotated through $45^{\circ}$ in the anti-clockwise direction about the point of trisection of $AB$ which is nearer to $B$. The equation of the line in the new position is:

$A$ line passing through the point $A(2, 3)$ makes an angle of $45^{\circ}$ with the $x$-axis and intersects the line $x + y + 1 = 0$ at point $P$. Find the distance $AP$.

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

The equation of a line,whose perpendicular distance from the origin is $7$ units and the angle,which the perpendicular to the line from the origin makes,is $120^{\circ}$ with the positive $X$-axis,is