Q4:

Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).

Given:

Let the two points be $A(-3, 10)$ and $B(6, -8)$. Let the point that divides the line segment $AB$ be $P(-1, 6)$.

To Find:

The ratio $m_1 : m_2$ in which the point $P$ divides the line segment $AB$.

Step 1: Applying the Section Formula

The Section Formula states that if a point $P(x, y)$ divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m_1 : m_2$, then the coordinates of $P$ are given by:

$x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}$ and $y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2}$

Let the ratio be $k : 1$. Thus, $m_1 = k$ and $m_2 = 1$.

Step 2: Substituting the Given Values

Here, $(x_1, y_1) = (-3, 10)$, $(x_2, y_2) = (6, -8)$, and $(x, y) = (-1, 6)$.

Using the x-coordinate formula:

$-1 = \frac{k(6) + 1(-3)}{k + 1}$

Step 3: Solving for k

Multiply both sides by $(k + 1)$:

$-1(k + 1) = 6k - 3$

Expand the left side:

$-k - 1 = 6k - 3$

Rearrange the terms to isolate $k$:

$-1 + 3 = 6k + k$

$2 = 7k$

$k = \frac{2}{7}$

Step 4: Verification using the y-coordinate

To ensure consistency, we check the y-coordinate using $k = \frac{2}{7}$:

$y = \frac{k(y_2) + 1(y_1)}{k + 1} = \frac{\frac{2}{7}(-8) + 1(10)}{\frac{2}{7} + 1}$

$y = \frac{-\frac{16}{7} + 10}{\frac{2+7}{7}} = \frac{\frac{-16 + 70}{7}}{\frac{9}{7}}$

$y = \frac{54}{7} \times \frac{7}{9} = \frac{54}{9} = 6$

Since the calculated y-coordinate matches the given y-coordinate of point $P$, the ratio is correct.

Final Answer: The point (-1, 6) divides the line segment joining (-3, 10) and (6, -8) in the ratio 2 : 7.