Question

In what ratio does the point (-4, 6) divide the line segment joining the points A(-6, 10) and B(3, -8)?

Hint:

Always verify your ratio with the y-coordinate: $\frac{2(-8) + 7(10)}{2+7} = \frac{-16+70}{9} = \frac{54}{9} = 6$. It matches!

Answer 1

Step 1: Assume the required ratio.
Let the point \(P(-4, 6)\) divide the line segment joining \(A(-6, 10)\) and \(B(3, -8)\) in the ratio \(m:n\). This means point \(P\) divides \(AB\) internally in the ratio \(m:n\).

Step 2: Use the section formula.
According to the section formula, if a point \(P(x, y)\) divides the line joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\), then
\(x = \frac{mx_2 + nx_1}{m+n}\) and \(y = \frac{my_2 + ny_1}{m+n}\).

Here,
\(A(-6,10)\), \(B(3,-8)\), and \(P(-4,6)\).

Step 3: Substitute values in the x-coordinate formula.
\(-4 = \frac{3m + (-6)n}{m+n}\)
\(-4(m+n) = 3m - 6n\)
\(-4m - 4n = 3m - 6n\)
\(2n = 7m\)

Step 4: Simplify the ratio.
From \(2n = 7m\), we get
\(m:n = 2:7\).

Step 5: Verify using the y-coordinate.
\(6 = \frac{-8m + 10n}{m+n}\)
\(6(m+n) = -8m + 10n\)
\(6m + 6n = -8m + 10n\)
\(14m = 4n\)
\(m:n = 2:7\).

Final Answer:
The point \( (-4, 6) \) divides the line segment joining \(A(-6,10)\) and \(B(3,-8)\) in the ratio 2 : 7.