Question

The midpoint of a line joining the points A (1,2a+1) and B(3b,3) is (5,-4).Find the values of a and b.

Answer 1

Given

• The midpoint of a line joining the points A (1,2a+1) and B(3b,3) is (5,-4)

To Find

• values of a and b

Concept used

• $\bf \: mid - point \: formula \\ \\ \bf \: x = \frac{ x_{1} + x_{2}}{2} \: and \: y = \frac{ y_{1} + y_{2} }{2}$

Solution

For absecissa

$\bf \: x = \dfrac{ x_{1} + x_{2} }{2} \\ \\ \implies \bf \: 5 = \frac{ 1 + 3b }{2} \\ \\ \implies \bf10 = 1 + 3b \\ \\ \implies \bf 9 = 3b \: \: \: \: \: \: \: \: \: \: \\ \\ \implies \underline{ \boxed{ \huge{ \bf \: b = 3}}}$

For ordinate

$\bf \: y = \dfrac{ y_{1} + y_{2} }{2} \\ \\ \implies \bf \: y = \frac{2a + 1 + 3}{2} \\ \\ \implies \bf \: - 4= \dfrac{2a + 4}{2} \: \: \: \: \: \\ \\ \implies \bf \: - 8 = 2a + 4 \: \: \: \: \: \: \\ \\ \bf \implies - 12 = 2a \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \underline{ \boxed{ \bf \huge \: a = - 6}}$

So the the value of a=-6 and b=3