If (x, y) be on the line joining the two points (1, -3) and (-4, 2), prove that x+y+2=0.

Question

Philoid · Accepted Answer

As the point (x, y) lies on the line joining the points (1, −3) and (−4, 2), it means that the three points are collinear.

Now, the condition of collinearity is: The area of the triangle formed by three points is 0.

Note: Area of the triangle having vertices (x₁,y₁), (x₂,y₂) and (x₃,y₃) is given as:
$Area( \triangle)= \frac{1}{2} |x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$
Now, for the three points to be colinear,

$Area( \triangle) = 0$

$\frac{1}{2} |x(-3 -2) + 1(2 -y) - 4(y + 3)| = 0$

-5x + 2 –y -4y -12 = 0

-5 x -5y -10 = 0

Taking "-5" common from the equation we get,

⇒ -5(x+y+2)=0

⇒ (x+y+2)

Hence proved, (x+y+2)=0