Coordinate Geometry

1 mark

The x-intercept of a line is the x-coordinate of the point where the line meets the x-axis, i.e. the value of $x$ when $y=0$.
The y-intercept of a line is the y-coordinate of the point where the line meets the y-axis, i.e. the value of $y$ when $x=0$.

1 mark

Let a straight line make x-intercept $a$ and y-intercept $b$. Then it passes through the points
$A(a,0)\quad \text{and}\quad B(0,b).$

Slope of the line, $m=\frac{b-0}{0-a}=-\frac{b}{a}.$

Using the point-slope form through $A(a,0)$, $y-0=-\frac{b}{a}(x-a).$

Multiplying by $a$, $ay=-b(x-a)$ $ay=-bx+ab$ $bx+ay=ab.$

Dividing both sides by $ab$, $\frac{x}{a}+\frac{y}{b}=1.$

Hence, the equation of the line in intercept form is
$\boxed{\frac{x}{a}+\frac{y}{b}=1}.$

4 marks

Let the line make equal and opposite intercepts on the axes.
So, if x-intercept is $a$, then y-intercept is $-a$.

Using the intercept form, $\frac{x}{a}+\frac{y}{-a}=1.$

$\frac{x}{a}-\frac{y}{a}=1.$

Multiplying by $a$, $x-y=a. \qquad \text{......(i)}$

The line passes through $(4,3)$. Substituting $x=4,\ y=3$ in (i), $4-3=a$ $\therefore a=1.$

Putting $a=1$ in (i), $x-y=1.$

Therefore, the required equation of the straight line is
$\boxed{x-y=1}$
or,
$\boxed{x-y-1=0}.$

Its intercepts are $1$ on the x-axis and $-1$ on the y-axis, which are equal and opposite.