Q.If $p$ is the length of the perpendicular from the origin to the line whose intercepts on the axes are $a$ and $b$, then show that $\dfrac{1}{p^2} = \dfrac{1}{a^2} + \dfrac{1}{b^2}$.

The general (linear) equation of a straight line is $ax+by+c=0$, where $a,b$ are not both zero; the set of solutions of any such equation is a straight line in the plane. Because dividing through by $b$ (or $a$) removes one constant, every line's equation genuinely contains only two independent arbitrary constants — so exactly two independent pieces of information (two points, or a point and a slope, or two intercepts, etc.) are enough to pin a line down uniquely.

Slope. The angle of inclination $\theta$ of a line is the angle it makes with the $x$-axis, measured counter-clockwise; the slope $m=\tan\theta$ (undefined when $\theta=\pi/2$, i.e. for a vertical line). Equivalently, through two points $(x_1,y_1),(x_2,y_2)$ with $x_1\ne x_2$, $m=\dfrac{y_2-y_1}{x_2-x_1}$; from the general form, $m=-a/b$ ($b\ne0$). Three points are collinear exactly when the slope of any one pair equals the slope of another pair sharing a point.

Intercepts. The $x$-intercept is where a line meets the $x$-axis ($y=0$); the $y$-intercept is where it meets the $y$-axis ($x=0$). ($x=0$ is itself the equation of the $y$-axis; $y=0$ is the equation of the $x$-axis.)

The six forms (two conditions each, all interconvertible by algebra):