# Position of two Points relative to a Line

The points P (x_{1}, y_{1}) and Q (x_{2} , y_{2})
lie on the same side or on opposite sides of the line a x +b y +c = 0
according as the expressions a x_{1} + b y_{1} +c and a x_{2}
+ by_{2} +c have same sign or opposite signs.

## Illustrative Examples

### Example

The sides of a triangle are given by the equations 3 x +4 y =
10, 4 x -3 y = 5 and 7 x +y +10 = 0. Show that the origin lies with in
the triangle.

.

### Solution

The given lines are

3 x +4 y -10 = 0 ...(i)

4 x -3 y -5 = 0 ...(ii)

and 7 x + y +10 = 0 ...(iii)

Let ABC be the triangle formed by these lines.

Solving these equations simultaneously, taking two at a time, the vertices of the triangle are

A (-2, 4), B (2, 1) and C (-1, -3)

On substituting (-2, 4) in L.H.S. of (ii), we get -8 -12 -5 = -25

and substituting (0, 0) in L.H.S. of (ii), we get 0 -0 -5 = -5

Since both have same sign, therefore, origin and A lie on the same side of BC.

Similarly origin and B lie on the same side of CA; and origin and C lie on the
same side of AB. (Please check it)

From these results, it follows that the origin lies with in the triangle
formed by the given lines.

## Exercise

- Are the points (2, 3) and (-1, 5) on the same side or on opposite sides of the line y = 2 x +5.
- Show that the points (3, 5) and (-3, -2)lie on the same side of the line 3 x = 7 y + 8.
- Which of the points (1, 1), (-1, 2), (2, 3) and (-3, 0) lie on the same side of the line 4 x +3 y = 5 on which the origin lies?
- Find by calculation whether the points (13, 8), (26, -4) lie in the same, adjacent or opposite angles formed by the straight line 5 x + 6 y -112 = 0 and 10 x +11 y - 217 = 0.
- Prove that the point P (x
_{1}, y_{1}) and the origin lie on the same side or on opposite sides of the line ax +by +c = 0 according as ax_{1}+ by_{1}+ c and c have same sign or opposite signs.

## Answers

**1.**Opposite sides

**3.**(-1, 2) and (-3, 0)

**4.**Opposite