If the lines p₁ x + q₁ y = 1, p₂ x + q₂ y = 1 and p₃ x + q₃ y = 1 be concurrent, show that the points (p₁, q₁), (p₂, q₂) and (p₃, q₃) are collinear.

Open in App

Solution

The given lines can be written as follows:

p₁ x + q₁ y $-$ 1 = 0 ... (1)

p₂ x + q₂ y $-$ 1 = 0 ... (2)

p₃ x + q₃ y $-$ 1 = 0 ... (3)

It is given that the three lines are concurrent.

$∴ |\begin{matrix} p_{1} & q_{1} & - 1 \\ p_{2} & q_{2} & - 1 \\ p_{3} & q_{3} & - 1 \end{matrix}| = 0 \Rightarrow - |\begin{matrix} p_{1} & q_{1} & 1 \\ p_{2} & q_{2} & 1 \\ p_{3} & q_{3} & 1 \end{matrix}| = 0 \Rightarrow |\begin{matrix} p_{1} & q_{1} & 1 \\ p_{2} & q_{2} & 1 \\ p_{3} & q_{3} & 1 \end{matrix}| = 0$

This is the condition for the collinearity of the three points, (p₁, q₁), (p₂, q₂) and (p₃, q₃).

Suggest Corrections

Similar questions

If the lines $p_{1} x + q_{1} y = 1, p_{2} x + q_{2} y = 1$ and $p_{3} x + q_{3} y = 1$ be concurrent, show that the points $(p_{1}, q_{1}), (p_{2}, q_{2})$ and $p_{3}, q_{3})$ are coolinear.

Q. If the lines

p_{1} x + q_{1} y = 1, p_{2} x + q_{2} y = 1

and

p_{3} x + q_{3} y = 1

be concurrent, then the points

(p_{1}, q_{1}), (p_{2}, q_{2})

and

(p_{3}, q_{3})

Q. If the lines

p_{1} x + q_{1} y = 1, p_{2} x + q_{2} y = 1

and

p_{3} x + q_{3} y = 1

be concurrent, then the points

(p_{1}, q_{1}) (p_{2}, q_{2}) (p_{3}, q_{3})

Q. Let

p (x) = a^{2} + b x, q (x) = l x^{2} + m x + n

. If

p (1) - q (1) = 0, p (2) - q (2) = 1

and

p (3) - q (3) = 4

, then

p (4) - q (4)

equals

Q. Let

p_{1} < p_{2} < p_{3} < p_{4}

and

q_{1} < q_{2} < q_{3} < q_{4}

be two sets of prime numbers such that

p_{4} - p_{1} = 8

and

q_{4} - q_{1} = 8

. Suppose,

p_{1} > 5

and

q_{1} > 5

. Prove that

30

divides

p_{1} - q_{1}

Join BYJU'S Learning Program

If the lines p1 x + q1 y = 1, p2 x + q2 y = 1 and p3 x + q3 y = 1 be concurrent, show that the points (p1, q1), (p2, q2) and (p3, q3) are collinear.

If the lines p₁ x + q₁ y = 1, p₂ x + q₂ y = 1 and p₃ x + q₃ y = 1 be concurrent, show that the points (p₁, q₁), (p₂, q₂) and (p₃, q₃) are collinear.