If (a, b) is the mid-point of the line segment joining the points A (10, –6) and B (k, 4) and a – 2b = 18, find the value of k and the distance AB

Solution:

Given, (a, b) is the midpoint of the line segment joining the points A(10, -6) and B(k, 4).

Also, a - 2b = 18 -------------- (1)

We have to find the value of k and the distance AB

The coordinates of the mid-point of the line segment joining the points P (x₁ , y₁) and Q (x₂ , y₂) are [(x₁ + x₂)/2, (y₁ + y₂)/2]

Here, (x₁ , y₁) = (10, -6) and (x₂ , y₂) = (k, 4)

Midpoint of AB, [(10 + k)/2, (-6 + 4)/2] = (a, b)

[(10 + k)/2, -2/2] = (a, b)

Now, (10 + k)/2 = a

10 + k = 2a

k = 2a - 10 ---------------- (2)

Also, b = -2/2

b = -1

Put b = -1 in (1)

a - 2(-1) = 18

a + 2 = 18

a = 18 - 2

a = 16

Put a = 16 in (2)

k = 2(16) - 10

k = 32 - 10

k = 22

Therefore, the value of k is 22.

The distance between two points P (x₁ , y₁) and Q (x₂ , y₂) is

√[(x₂ - x₁)² + (y₂ - y₁)²]

Distance between A(10, -6) and B(22, 4) = √[(22 - 10)² + (4 - (-6))²]

= √[(12)² + (10)²]

= √(144 + 100)

= √244

= 2√61

Therefore, the distance between A and B is 2√61 units.

✦ Try This: If P(1/2 ,4) is the midpoint of the line segment joining the points Q(-6, 5) and fl(-2, 3), then the value of a is

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 7

NCERT Exemplar Class 10 Maths Exercise 7.3 Problem 13

If (a, b) is the mid-point of the line segment joining the points A (10, -6) and B (k, 4) and a - 2b = 18, find the value of k and the distance AB

Summary:

If (a, b) is the midpoint of the line segment joining the points A (10, - 6) and B (k, 4) and a - 2b = 18, the value of k is 22 and the distance AB is 2√61 units

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