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(x_{1},y_{1}) and Q(x_{2},y_{2}) 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(mx_{2}+nx_{1})+b(my_{2}+ny_{1})+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 ax_{1}+by_{1}+c and ax_{2}+by_{2}+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 ax_{1}+by_{1}+c and ax_{2}+by_{2}+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

- Two given points P(x
_{1}, y_{1}) and Q(x_{2}, y_{2}) will lie on the**same**side of the line ax+by+c=0 if ax_{1}+by_{1}+c and ax_{2}+by_{2}+c will have**same**signs. - On the other hand, P(x
_{1}, y_{1}) and Q(x_{2}, y_{2}) will lie on the**opposite**sides of the line ax+by+c=0 if ax_{1}+by_{1}+c and ax_{2}+by_{2}+c will have**opposite**signs.

See you in the next lesson with a few examples.

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