Courses
Courses for Kids
Free study material
Offline Centres
More

If the points $(p,q),(m,n)$ and $\left( {p - m,q - n} \right)$ are collinear, show that $pn = qm$

seo-qna
Last updated date: 28th Feb 2024
Total views: 408.9k
Views today: 7.08k
IVSAT 2024
Answer
VerifiedVerified
408.9k+ views
Hint: If three points are collinear then the area of the triangle formed by those three points will be 0.
If $\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right),\left( {{x_3},{y_3}} \right)$are three points, then the area of triangle formed with these points is
Area of triangle = $\frac{1}{2}\left( {{x_1}({y_2} - {y_3}) + {x_2}({y_3} - {y_1}) + {x_3}({y_1} - {y_2})} \right)$--- (1)

Given points $(p,q),(m,n)$ and $\left( {p - m,q - n} \right)$ are collinear. Therefore, from equation (1)
The area of triangle = $$\frac{1}{2}\left( {p(n - q + n) + m(q - n - q) + (p - m)(q - n)} \right)$$
$ \Rightarrow 0 = \frac{1}{2}\left( {p(2n - q) - mn + pq - pn - qm + mn} \right)$
$ \Rightarrow 0 = \frac{1}{2}(2pn - pq + pq - pn - qm)$
$ \Rightarrow 0 = pn - qm$
$ \Rightarrow pn = qm$
Hence proved.
Note: Three points A, B and C are said to be collinear means they are lying on the same straight line. Then the area formed by points ABC will be equal to zero. We can also check collinearity by using distance between points that is AB+BC=AC

Recently Updated Pages