Question

$\huge \pink hello$

please help me in the above attachment... ♥

St lines

Answer 1

As these three lines are Concurrent : they all intersect at a single point

⇒ The Given three lines have Common point of Intersection

Let us find the Point of intersection of any two lines :

1st line : y = m₁x + c₁

substituting the value of y in the 2nd line : y = m₂x + c₂

⇒ m₁x + c₁ = m₂x + c₂

⇒ x(m₁ - m₂) = c₂ - c₁

⇒ $x = \frac{c_{2} - c_{1}}{m_1 - m_2}$

⇒ $y = \frac{m_1(c_2 - c_1)}{m_1 - m_2} + c_1$

substituting the x and y values in 3rd line : y = m₃x + c₃

⇒ $\frac{m_1(c_2 - c_1)}{m_1 - m_2} + c_1 = \frac{m_3(c_2 - c_1)}{m_1 - m_2} + c_3$

⇒ $\frac{m_1(c_2 - c_1)}{m_1 - m_2} = \frac{m_3(c_2 - c_1)}{m_1 - m_2} + c_3 - c_1$

⇒ m₁(c₂ - c₁) = m₃(c₂ - c₁) + (m₁ - m₂)(c₃ - c₁)

⇒ m₃(c₂ - c₁) + m₁(c₃ - c₁ - c₂ + c₁) - m₂(c₃ - c₁) = 0

⇒ m₃(c₂ - c₁) + m₁(c₃ - c₂) - m₂(c₃ - c₁) = 0

⇒ m₃(c₁ - c₂) + m₁(c₂ - c₃) + m₂(c₃ - c₁) = 0

Answer 2

Answer:

As these three lines are Concurrent : they all intersect at a single point

⇒ The Given three lines have Common point of Intersection

Let us find the Point of intersection of any two lines :

1st line : y = m₁x + c₁

substituting the value of y in the 2nd line : y = m₂x + c₂

⇒ m₁x + c₁ = m₂x + c₂

⇒ x(m₁ - m₂) = c₂ - c₁

⇒ x = \frac{c_{2} - c_{1}}{m_1 - m_2}x=

m

1

−m

2

c

2

−c

1

⇒ y = \frac{m_1(c_2 - c_1)}{m_1 - m_2} + c_1y=

m

1

−m

2

m

1

(c

2

−c

1

)

+c

1

substituting the x and y values in 3rd line : y = m₃x + c₃

⇒ \frac{m_1(c_2 - c_1)}{m_1 - m_2} + c_1 = \frac{m_3(c_2 - c_1)}{m_1 - m_2} + c_3

m

1

−m

2

m

1

(c

2

−c

1

)

+c

1

=

m

1

−m

2

m

3

(c

2

−c

1

)

+c

3

⇒ \frac{m_1(c_2 - c_1)}{m_1 - m_2} = \frac{m_3(c_2 - c_1)}{m_1 - m_2} + c_3 - c_1

m

1

−m

2

m

1

(c

2

−c

1

)

=

m

1

−m

2

m

3

(c

2

−c

1

)

+c

3

−c

1

⇒ m₁(c₂ - c₁) = m₃(c₂ - c₁) + (m₁ - m₂)(c₃ - c₁)

⇒ m₃(c₂ - c₁) + m₁(c₃ - c₁ - c₂ + c₁) - m₂(c₃ - c₁) = 0

⇒ m₃(c₂ - c₁) + m₁(c₃ - c₂) - m₂(c₃ - c₁) = 0

⇒ m₃(c₁ - c₂) + m₁(c₂ - c₃) + m₂(c₃ - c₁) = 0