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In each of the following examples find the co-ordinates of point A which divides segment PQ in the ratio a: b. P(2, 6), Q(-4, 1), a : b = 1 : 2

2. In each of the following examples find the co-ordinates of point A which divides segment PQ in the ratio a: b. P(2, 6), Q(-4, 1), a : b = 1 : 2

Answer:-

Let the coordinates of point A be (x, y).

P(2, 6), Q(–4, 1), a : b = 1 : 2
Using section formula

\[x = \frac{1 \times \left( - 4 \right) + 2 \times 2}{1 + 2} = \frac{- 4 + 4}{3} = 0\]

\[y = \frac{1 \times 1 + 2 \times 6}{1 + 2} = \frac{1 + 12}{3} = \frac{13}{3}\]

\[\left( x, y \right) = \left( 0, \frac{13}{3} \right)\]

Explanation:-

To find the coordinates of point A which divides segment PQ in the ratio a:b, we use the section formula.

The section formula states that if we have two points in a 2D plane, P(x1, y1) and Q(x2, y2), and we want to find the coordinates of a point A that divides the segment PQ in the ratio a:b, then the coordinates of A are given by:

A = ((bx2 + ax1)/(a + b), (by2 + ay1)/(a + b))

In this case, we have P(2, 6), Q(-4, 1), and a:b = 1:2. This means that the segment PQ is divided into three parts, with A being the point that divides it into a part of length a and a part of length b, where b is twice as long as a.

To find the coordinates of A, we substitute the given values into the section formula:

A = ((2*(-4) + 12)/(1 + 2), (21 + 1*6)/(1 + 2))

Simplifying this expression, we get:

A = (-2/3, 5)

Therefore, the coordinates of point A are (-2/3, 5), and it divides the segment PQ in the ratio 1:2.

Chapter 5. Co-ordinate Geometry – Practice Set 5.2 (Page 115)