# Equation of a line (Coordinate geometry)

• Feb 3rd 2010, 02:04 AM
snigdha
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).
• Feb 3rd 2010, 02:53 AM
Hello snigdha
Quote:

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 $\displaystyle m$. Then its equation is:
$\displaystyle y-1=m(x-3)$
Simplify this, and then find the coordinates of the intercepts. They are:
$\displaystyle \left(\frac{m-3}{m},0\right)$ and $\displaystyle (0, 3-m)$
Then, using $\displaystyle \tfrac12$ base x height, write down an expression to represent the area of the triangle, and equate this to $\displaystyle \pm 6$. (You need the $\displaystyle \pm$ sign, because you might have negative distances, and hence a negative area.

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

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