If the lines p1x + q1y = 1, p2x + q2y = 1 and p3x + q3y = 1 be concurrent, show that the points (p1, q1), (p2, q2) and (p3, q3) are collinear.

Home » If the lines p1x + q1y = 1, p2x + q2y = 1 and p3x + q3y = 1 be concurrent, show that the points (p1, q1), (p2, q2) and (p3, q3) are collinear.

If the lines p1x + q1y = 1, p2x + q2y = 1 and p3x + q3y = 1 be concurrent, show that the points (p1, q1), (p2, q2) and (p3, q3) are collinear.

Given:

${{p}_{1}}x\text{ }+\text{ }{{q}_{1}}y\text{ }=\text{ }1$

${{p}_{2}}x\text{ }+\text{ }{{q}_{2}}y\text{ }=\text{ }1$

and ,

${{p}_{3}}x\text{ }+\text{ }{{q}_{3}}y\text{ }=\text{ }1$

The given lines can be written as follows:

${{p}_{1}}~x\text{ }+\text{ }{{q}_{1}}~y\text{ }\text{ }1\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)$

${{p}_{2}}~x\text{ }+\text{ }{{q}_{2}}~y\text{ }\text{ }1\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 2 \right)$

and,

${{p}_{3}}~x\text{ }+\text{ }{{q}_{3}}~y\text{ }\text{ }1\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 3 \right)$

since, the three lines are concurrent.

Now, consider the following determinant:

Hence proved, the given three points,

(p₁, q₁), (p₂, q₂) and (p₃, q₃) are collinear.

i

Sign up now and study all subjects for free