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.

Points relative to a line in the Cartesian plane

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.

Points relative to a line in the Cartesian plane

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.


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