Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).
Step-by-Step Solution
Key Concept: Use the section formula (internal division) which states that if a point P(x, y) divides the line segment joining A(x₁, y₁) and B(x₂, y₂) in the ratio m:n (i.e., AP : PB = m : n), then \[ x = \frac{n x_1 + m x_2}{m+n}, \qquad y = \frac{n y_1 + m y_2}{m+n}. \] Solve for the ratio m:n using the given coordinates.
1. Identify the given points\
\[ A(-3,\,10), \quad B(6,\,-8), \quad P(-1,\,6). \]\
2. Assume the required ratio\
Let \(AP : PB = m : n\). Here \(m\) corresponds to the part from \(A\) to \(P\) and \(n\) to the part from \(P\) to \(B\).\
3. Write the section‑formula equations\
\[ -1 = \frac{n(-3) + m(6)}{m+n}, \qquad 6 = \frac{n(10) + m(-8)}{m+n}. \]\
4. Clear the denominators\
\[ - (m+n) = -3n + 6m \quad\Rightarrow\quad m+n = 3n - 6m \quad\text{(i)} \]
\[ 6(m+n) = 10n - 8m \quad\Rightarrow\quad 6m + 6n = 10n - 8m \quad\text{(ii)} \]
5. Solve the simultaneous equations\
From (i): \(m + n = 3n - 6m \Rightarrow 7m = 2n \Rightarrow n = \frac{7}{2}m\).\
Substitute \(n = \frac{7}{2}m\) in (ii):\
\(6m + 6\left(\frac{7}{2}m\right) = 10\left(\frac{7}{2}m\right) - 8m\) which is satisfied, confirming the relation.\
6. Find the simplest integer ratio\
\(n = \frac{7}{2}m \Rightarrow \frac{m}{n} = \frac{2}{7}.\)\
Hence \(AP : PB = m : n = 2 : 7\).\
7. Verification (optional)\
Using the ratio 2:7, the coordinates of the dividing point are\
\[ x = \frac{7(-3) + 2(6)}{2+7} = \frac{-21 + 12}{9} = -1, \]
\[ y = \frac{7(10) + 2(-8)}{9} = \frac{70 - 16}{9} = 6, \]
which matches the given point \((-1,6)\).
Correct Answer: 2 : 7