Question

A point P divides the line segment joining the points A(-3, 5) and B(7, -4) in a certain ratio. If the point P lies on the line y = 2x, then find the ratio AP : PB and coordinates of point P.

Hint:

Using \(k : 1\) as the ratio is always easier than using \(m_1 : m_2\) because it reduces the problem to solving a single-variable equation.

Answer 1

Step 1: Understanding the Question:
We are given two points \(A(-3, 5)\) and \(B(7, -4)\).
A point \(P\) lies on the line segment \(AB\) and divides it in a ratio.
This point \(P\) also lies on the line represented by \(y = 2x\).
We need to determine the ratio of division \(AP : PB\) and the coordinates of \(P\).

Step 2: Key Formula or Approach:
1. Let the point \(P\) divide the segment \(AB\) in the ratio \(k : 1\).
2. Use the Section Formula to express the coordinates of \(P(x, y)\) in terms of \(k\):
\[ x = \frac{k x_2 + x_1}{k + 1}, \quad y = \frac{k y_2 + y_1}{k + 1} \] 3. Since \(P\) lies on the line \(y = 2x\), substitute these coordinate expressions into the line's equation and solve for \(k\).

Step 3: Detailed Explanation:
1. Write down the coordinates:
- \(A(x_1, y_1) = (-3, 5)\)
- \(B(x_2, y_2) = (7, -4)\)
2. Apply the Section Formula with ratio \(k : 1\):
\[ x = \frac{7k - 3}{k + 1} \] \[ y = \frac{-4k + 5}{k + 1} \] 3. Substitute \(x\) and \(y\) into the equation of the line \(y = 2x\):
\[ \frac{-4k + 5}{k + 1} = 2 \left( \frac{7k - 3}{k + 1} \right) \] 4. Multiply both sides by \(k + 1\) (assuming \(k \neq -1\)):
\[ -4k + 5 = 2(7k - 3) \] \[ -4k + 5 = 14k - 6 \] 5. Group the \(k\) terms on one side:
\[ 14k + 4k = 5 + 6 \] \[ 18k = 11 \implies k = \frac{11}{18} \] So, the division ratio \(AP : PB\) is \(11 : 18\).
6. Now, calculate the coordinates of \(P\) by substituting \(k = \frac{11}{18}\) back:
\[ x = \frac{7\left(\frac{11}{18}\right) - 3}{\frac{11}{18} + 1} = \frac{\frac{77 - 54}{18}}{\frac{11 + 18}{18}} = \frac{23}{29} \] \[ y = 2x = 2\left(\frac{23}{29}\right) = \frac{46}{29} \] 7. The coordinates of point \(P\) are \(\left(\frac{23}{29}, \frac{46}{29}\right)\).

Step 4: Final Answer:
The ratio \(AP : PB\) is \(11 : 18\) and the coordinates of point \(P\) are \(\left(\frac{23}{29}, \frac{46}{29}\right)\).