Question

If R (x,y) is a point on the line segment joining the point p (a,b) and q (b,a) then prove that x+y=a+b.

Answer 1

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Step-by-step explanation:

Answer 2

Given:

R (x,y) is a point on the line segment joining the point p (a,b) and q (b, a).

To prove:

x+y=a+b.

Solution:

As given, R (x,y) is a point on the line segment joining the point p (a,b) and q (b, a), hence, all the given three points are collinear.

As we know that, if three points are collinear, no triangle can be formed. Hence, the area of the triangle will be zero.

So, we have

The area of the triangle formed by three points (x, y), (p, q) and (m, n) is =

$\: \frac{1}{2} [ x(q - n) + p(y - n) + m(y - q)] = 0$

So, using three points p (a, b), R (x, y) and q (b, a), we have

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

$ay - {a}^{2} + xa - xb + {b}^{2} - by = 0$

${b}^{2} - {a}^{2} + y(a - b) + x(a - b) = 0$

$y(a - b) + x(a - b) = {a}^{2} - {b}^{2}$

$(a - b) (x + y) = (a - b) (a + b)$

$[using \: identity \:{a}^{2} - {b}^{2} = (a - b) (a + b)]$

$(x + y) = (a + b)$

[cancelled (a - b) both sides]

Hence proved x+y=a+b.