# Position of a point relative to a line

Hello. This lesson will deal with a relatively easy concept, as compared to the recently covered parametric equation, to let you relax a bit.

The focus will be on determination of relative positions of two (or more) points with respect to a given line. That is, whether two given points lie on the same side of a given line, or opposite.

Let L be a given line ax + by + c = 0, and P(x1,y1) and Q(x2,y2) be two points.

Let R(x, y) be a point on the line L, such that P, Q and R are collinear, and let this point R divide PQ in the ratio m:n.

Now the coordinates of R (using section formula) will be $$(\frac{mx_2 + nx_1}{m+n},\frac{my_2+ny_1}{m+n})$$

Since the point also lies on the line ax + by + c = 0, its coordinates will satisfy the equation.

Therefore, we have a(mx2+nx1)+b(my2+ny1)+c(m+n)=0, or $$\frac{m}{n}=-\frac{ax_1+by_1+c}{ax_2+by_2+c}$$ [1]

Now here’s how we’ll determine the relative position of the two points.

If ax1+by1+c and ax2+by2+c are of the opposite sign (i.e. one positive and other negative), then RHS of [1] will be positive which makes the m/n positive, implying that R divides PQ in the ratio m:n internally.

And this can only happen when R lies between P and Q, or P and Q lie on the opposite sides of the given line.

On the other hand, if ax1+by1+c and ax2+by2+c are of the same sign (i.e. both positive or both negative), then RHS of [1] will be negative, which makes m/n negative, implying that R divides PQ in the ratio m:n externally.

And this can only happen if R lies outside the line segment PQ, or P and Q lie on the same side of R, or the given line.

That’s all there is to it!

There might be a case when there isn’t any point R on the given line such that P, Q and R are collinear. This will happen when PQ is parallel to the given line, which implies P and Q lie on the same side of the line. (You needn’t worry about this case separately.)

## Lesson Summary

1. Two given points P(x1, y1) and Q(x2, y2) will lie on the same side of the line ax+by+c=0 if ax1+by1+c and ax2+by2+c will have same signs.
2. On the other hand, P(x1, y1) and Q(x2, y2) will lie on the opposite sides of the line ax+by+c=0 if ax1+by1+c and ax2+by2+c will have opposite signs.

See you in the next lesson with a few examples.