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Mathematics

If P(-b, 9a - 2) divides the line segment joining the points A(-3, 3a + 1) and B(5, 8a) in the ratio 3 : 1, find the values of a and b.

Section Formula

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Answer

By section-formula,

x=m1x2+m2x1m1+m2x = \dfrac{m1x2 + m2x1}{m1 + m2}

Substituting values we get,

⇒−b=3×5+1×−33+1⇒−b=15−34⇒−b=124=3⇒b=−3y=m1y2+m2y1m1+m2\Rightarrow -b = \dfrac{3 \times 5 + 1 \times -3}{3 + 1} \\[1em] \Rightarrow -b = \dfrac{15 - 3}{4} \\[1em] \Rightarrow -b = \dfrac{12}{4} = 3 \\[1em] \Rightarrow b = -3 \\[1em] y = \dfrac{m1y2 + m2y1}{m1 + m2}

Substituting values we get,

⇒9a−2=3×8a+1×(3a+1)3+1⇒9a−2=24a+3a+14⇒9a−2=27a+14⇒36a−8=27a+1⇒9a=9⇒a=1.\Rightarrow 9a - 2 = \dfrac{3 \times 8a + 1 \times (3a + 1)}{3 + 1} \\[1em] \Rightarrow 9a - 2 = \dfrac{24a + 3a + 1}{4} \\[1em] \Rightarrow 9a - 2 = \dfrac{27a + 1}{4} \\[1em] \Rightarrow 36a - 8 = 27a + 1 \\[1em] \Rightarrow 9a = 9 \\[1em] \Rightarrow a = 1.

Hence, a = 1 and b = -3.

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