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Mathematics

Calculate the ratio in which the line joining A(-4, 2) and B(3, 6) is divided by point P(x, 3). Also, find (i) x (ii) length of AP.

Section Formula

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Answer

Let ratio be m1 : m2.

By section formula,

y=m1y2+m2y1m1+m2y = \dfrac{m1y2 + m2y1}{m1 + m2}

Substituting values we get,

3=m1×6+m2×2m1+m2⇒3m1+3m2=6m1+2m2⇒m2=3m1⇒m1m2=13.3 = \dfrac{m1 \times 6 + m2 \times 2}{m1 + m2} \\[1em] \Rightarrow 3m1 + 3m2 = 6m1 + 2m2 \\[1em] \Rightarrow m2 = 3m1 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{1}{3}.

m1 : m2 = 1 : 3.

(i) By section formula,

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

Substituting values we get,

⇒x=1×3+3×−41+3⇒x=3−124⇒x=−94.\Rightarrow x = \dfrac{1 \times 3 + 3 \times -4}{1 + 3} \\[1em] \Rightarrow x = \dfrac{3 - 12}{4} \\[1em] \Rightarrow x = -\dfrac{9}{4}.

Hence, x = −94-\dfrac{9}{4}.

(ii) Distance between two points = (x2−x1)2+(y2−y1)2\sqrt{(x2 - x1)^2 + (y2 - y1)^2}

AP=[−94−(−4)]2+[3−2]2=[−94+4]2+[1]2=[−9+164]2+1=[74]2+1=4916+1=49+1616=6516=654.AP = \sqrt{\Big[-\dfrac{9}{4} - (-4)\Big]^2 + [3 - 2]^2} \\[1em] = \sqrt{\Big[-\dfrac{9}{4} + 4\Big]^2 + [1]^2} \\[1em] = \sqrt{\Big[\dfrac{-9 + 16}{4}\Big]^2 + 1} \\[1em] = \sqrt{\Big[\dfrac{7}{4}\Big]^2 + 1} \\[1em] = \sqrt{\dfrac{49}{16} + 1} \\[1em] = \sqrt{\dfrac{49 + 16}{16}} \\[1em] = \sqrt{\dfrac{65}{16}} \\[1em] = \dfrac{\sqrt{65}}{4}.

Hence, AP = 654.\dfrac{\sqrt{65}}{4}.

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