If R( x , y ) is a point on the line segment joining the points P( a , b - Maths - Coordinate Geometry

Question

If R( x , y ) is a point on the line segment joining the points
P( a , b ) and Q( b , a ) , then prove that x+y =a+b

Lalit Mehra · Accepted Answer

Since R(x, y) is a point on the line segment joining the point, P(a, b) and Q(b, a).

∴ P(a, b), Q(b, a) and R(x, y)are the collinear.

⇒ Area of ΔPQR = 0

Area of triangle whose vertices are (x₁, y₁), (x₂, y₂) and (x₃, y₃) $is \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]$ .

$∴ \frac{1}{2} [a (a - y) + b (y - b) + x (b - a)] = 0 \Rightarrow a^{2} - a y + b y - b^{2} + x (b - a) = 0 \Rightarrow y (b - a) + x (b - a) = b^{2} - a^{2} \Rightarrow (x + y) (b - a) = (b - a) (b + a) \Rightarrow x + y = a + b$

If R( x , y ) is a point on the line segment joining the points P( a , b ) and Q( b , a ) , then prove that x+y =a+b