Question

If the three lines p1x + q1y = 1, p2x + q2y = 1 and p3x + q3y = 1 are concurrent, then the points (p1, q1 ), (p2, q2 ) and (p3, q3 ) are

Answer 1

$\large\underline{\sf{Solution-}}$

Given that, three lines

$\rm \: p_1x + q_1y = 1$

$\rm \: p_2x + q_2y = 1$

$\rm \: p_3x + q_3y = 1$

are concurrent.

The above 3 lines can be rewritten as

$\rm \: p_1x + q_1y - 1 = 0 - - - (1)$

$\rm \: p_2x + q_2y - 1 = 0 - - - (2)$

$\rm \: p_3x + q_3y - 1 = 0 - - - (3)$

Since, it is given that above three lines are concurrent.

$\rm\implies \: \: \begin{gathered}\sf \left | \begin{array}{ccc}p_1&q_1& - 1\\p_2&q_2& - 1\\p_3&q_3& - 1\end{array}\right | \end{gathered} = 0$

Take out (- 1) common from third column, we get

$\rm\implies \: \: - \begin{gathered}\sf \left | \begin{array}{ccc}p_1&q_1& 1\\p_2&q_2&1\\p_3&q_3&1\end{array}\right | \end{gathered} = 0$

$\rm\implies \: \: \begin{gathered}\sf \left | \begin{array}{ccc}p_1&q_1& 1\\p_2&q_2&1\\p_3&q_3&1\end{array}\right | \end{gathered} = 0$

$\rm\implies \: \: \dfrac{1}{2} \begin{gathered}\sf \left | \begin{array}{ccc}p_1&q_1& 1\\p_2&q_2&1\\p_3&q_3&1\end{array}\right | \end{gathered} = 0$

$\rm\implies \:(p_1,q_1), \:(p_2,q_2), \: (p_3,q_3)\: \: are \: collinear$

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Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to x - axis passes through the point (a, b) is y = b.

Equation of line parallel to y - axis passes through the point (a, b) is x = a.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

Equation of line which makes an intercept of c units on y axis and having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.

Answer 2

Question:-

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.

Answer:-

[Refer to the above attachment]

Hope you have satisfied. ⚘