# Thread: Equation of a line (Coordinate geometry)

1. ## Equation of a line (Coordinate geometry)

Find the equation of a line through (1,3) and forming with the axes a triangle of area 6 square units. (Give three solutions).

2. Hello snigdha
Originally Posted by snigdha
Find the equation of a line through (1,3) and forming with the axes a triangle of area 6 square units. (Give three solutions).
Suppose the line has gradient $m$. Then its equation is:
$y-1=m(x-3)$
Simplify this, and then find the coordinates of the intercepts. They are:
$\left(\frac{m-3}{m},0\right)$ and $(0, 3-m)$
Then, using $\tfrac12$ base x height, write down an expression to represent the area of the triangle, and equate this to $\pm 6$. (You need the $\pm$ sign, because you might have negative distances, and hence a negative area.

Re-write this as a pair of quadratic equations in $m$ (one equation for each of the two possible signs.)

Solve these to find the possible values of $m$, and hence the equations of the lines.
Spoiler:
The values of $m$ are $m = -3$, or $9\pm6\sqrt2$