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Mathematics

The mid-point of the line segment joining (4a, 2b - 3) and (-4, 3b) is (2, -2a). Find the values of a and b.

Section Formula

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Answer

By formula,

Mid-point (M) = (x1+x22,y1+y22)\Big(\dfrac{x1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

Substituting values we get,

⇒(2,−2a)=(4a+(−4)2,2b−3+3b2)∴2=4a−42 and −2a=5b−32⇒4a−4=4 and 5b−3=−4a⇒4a=8 and 5b=−4a+3⇒a=2 and 5b=−4(2)+3⇒a=2 and 5b=−5⇒a=2 and b=−1.\Rightarrow (2, -2a) = \Big(\dfrac{4a + (-4)}{2}, \dfrac{2b - 3 + 3b}{2}\Big) \\[1em] \therefore 2 = \dfrac{4a - 4}{2} \text{ and } -2a = \dfrac{5b - 3}{2} \\[1em] \Rightarrow 4a - 4 = 4 \text{ and } 5b - 3 = -4a \\[1em] \Rightarrow 4a = 8 \text{ and } 5b = -4a + 3 \\[1em] \Rightarrow a = 2 \text{ and } 5b = -4(2) + 3 \\[1em] \Rightarrow a = 2 \text{ and } 5b = -5 \\[1em] \Rightarrow a = 2 \text{ and } b = -1.

Hence, a = 2 and b = -1.

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