4.4 Q-4

Question Statement

Find the condition that the lines $y = m_1 x + c_1$, $y = m_2 x + c_2$, and $y = m_3 x + c_3$ are concurrent.

That is, determine the relationship between the slopes and intercepts of these lines for them to intersect at a single point.

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Background and Explanation

In geometry, three lines are said to be concurrent if they all meet at a single point. To solve for the condition of concurrency, we need to find the relationship between the constants $m_1, m_2, m_3$ (slopes) and $c_1, c_2, c_3$ (intercepts) of the three given lines.

The approach involves solving a system of equations for $x$ and $y$, the coordinates of the intersection point. The lines are represented by their general equations, and we aim to derive a condition under which all three equations are satisfied simultaneously.

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Solution

We are given three equations of straight lines:

1. $m_1 x - y + c_1 = 0$
2. $m_2 x - y + c_2 = 0$
3. $m_3 x - y + c_3 = 0$

Step 1: Solve for the intersection of the first two lines

From the first two equations, we can solve for $x$ and $y$.

Starting with the first and second equations:

$$m_1 x - y + c_1 = 0 \tag{1}$$ $$m_2 x - y + c_2 = 0 \tag{2}$$

Subtract equation (2) from equation (1):

$$(m_1 x - y + c_1) - (m_2 x - y + c_2) = 0$$

Simplifying:

$$(m_1 - m_2) x + (c_1 - c_2) = 0$$

Solving for $x$:

$$x = \frac{c_2 - c_1}{m_1 - m_2} \tag{3}$$

Step 2: Substitute into one of the original equations to find $y$

Now substitute the value of $x$ from equation (3) into either of the original equations (let’s use equation (1)):

$$m_1 \left( \frac{c_2 - c_1}{m_1 - m_2} \right) - y + c_1 = 0$$

Solving for $y$:

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

Thus, the coordinates of the intersection of the first two lines are:

$$x = \frac{c_2 - c_1}{m_1 - m_2}, \quad y = \frac{m_1 (c_2 - c_1)}{m_1 - m_2} + c_1$$

Step 3: Substitute these into the third equation

Now, substitute $x$ and $y$ into the third equation $m_3 x - y + c_3 = 0$:

$$m_3 \left( \frac{c_2 - c_1}{m_1 - m_2} \right) - \left( \frac{m_1 (c_2 - c_1)}{m_1 - m_2} + c_1 \right) + c_3 = 0$$

Simplify the equation to find the condition for concurrency:

$$m_3 \left( c_2 - c_1 \right) - m_2 c_1 - m_1 c_2 + m_2 c_3 - m_1 c_3 = 0$$

Rearranging terms:

$$m_3 (c_2 - c_1) + m_2 (c_3 - c_1) + m_1 (c_1 - c_3) = 0$$

Step 4: Final condition for concurrency

Thus, the condition for the three lines to be concurrent is:

$$(m_2 - m_1)(c_3 - c_1) = (m_3 - m_1)(c_2 - c_1)$$

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Key Formulas or Methods Used

• Equation of a line: $y = mx + c$
• Solving a system of linear equations
• Condition for three lines to be concurrent

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Summary of Steps

1. Write the equations of the three lines.
2. Solve for $x$ and $y$ using the first two lines.
3. Substitute $x$ and $y$ into the third line’s equation.
4. Simplify to find the condition for concurrency:

$$(m_2 - m_1)(c_3 - c_1) = (m_3 - m_1)(c_2 - c_1)$$