Question 16 of 18intermediate🔧 ApplyNumerical3 marks

If A and B are (-2, -2) and (2, -4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.

Correct Answer

The coordinates of P are (-2/7, -20/7).

Exercise: EXERCISE 7.2 | Q: 8 | (Page 13)
For More Understanding

Explanation

This question tests the application of the section formula. Since AP = 3/7 AB and P lies on segment AB, we can deduce that PB = AB - AP = 4/7 AB. Therefore, P divides AB internally in the ratio 3:4. The section formula from Example 8 in the context shows that when a point divides a line segment in ratio m:n, we use the formula ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)).

Solution Steps

  1. Step 1: Given AP = 3/7 AB. Since P lies on AB, we have PB = AB - AP = AB - 3/7 AB = 4/7 AB.

  2. Step 2: Therefore, the ratio AP : PB = 3/7 AB : 4/7 AB = 3 : 4. So P divides AB internally in the ratio 3:4.

  3. Step 3: Using the section formula, P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)) where m = 3, n = 4, A(-2, -2) and B(2, -4).

  4. Step 4: x-coordinate = (3(2) + 4(-2))/(3+4) = (6 - 8)/7 = -2/7

  5. Step 5: y-coordinate = (3(-4) + 4(-2))/(3+4) = (-12 - 8)/7 = -20/7

  6. Step 6: Therefore, the coordinates of P are (-2/7, -20/7).

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