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In what ratio is the line joining A(0, 3) and B (4, -1) divided by the x-axis ?

Write the co-ordinates of the point where AB intersects the x-axis.

Section Formula

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Answer

Let AB intersect x-axis at P. So, co-ordinates of P = (x, 0).

Let ratio be m1 : m2.

By section formula,

y=m1y2+m2y1m1+m20=m1×1+m2×3m1+m20=m1+3m2m1=3m2m1m2=31.y = \dfrac{m1y2 + m2y1}{m1 + m2} \\[1em] \Rightarrow 0 = \dfrac{m1 \times -1 + m2 \times 3}{m1 + m2} \\[1em] \Rightarrow 0 = -m1 + 3m2 \\[1em] \Rightarrow m1 = 3m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{3}{1}.

m1 : m2 = 3 : 1.

x=m1x2+m2x1m1+m2=3×4+1×03+1=12+04=124=3.x = \dfrac{m1x2 + m2x1}{m1 + m2} \\[1em] = \dfrac{3 \times 4 + 1 \times 0}{3 + 1} \\[1em] = \dfrac{12 + 0}{4} \\[1em] = \dfrac{12}{4} = 3. \\[1em]

P = (x, 0) = (3, 0).

Hence, co-ordinates of P = (3, 0) and ratio = 3 : 1.

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