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M and N are two points on the x-axis and y-axis respectively. P(3, 2) divides the line segment MN in the ratio 2 : 3. Find :

(i) the coordinates of M and N.

(ii) slope of the line MN.

Straight Line Eq

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Answer

(i) Let the coordinates of M and N be (x, 0) and (0, y).

By section formula the coordinates of P are,

=(m1x2+m2x1m1+m2,m1y2+m2y1m1+m2)=(2×0+3×x2+3,2×y+3×02+3)=(3x5,2y5)= \Big(\dfrac{m1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big) \\[1em] = \Big(\dfrac{2 \times 0 + 3 \times x}{2 + 3}, \dfrac{2 \times y + 3 \times 0}{2 + 3}\Big) \\[1em] = \Big(\dfrac{3x}{5}, \dfrac{2y}{5}\Big)

Given, P(3, 2). Comparing two values of P we get,

⇒3=3x5 and 2=2y5⇒3x=15 and 2y=10⇒x=5 and y=5.\Rightarrow 3 = \dfrac{3x}{5} \text{ and } 2 = \dfrac{2y}{5} \\[1em] \Rightarrow 3x = 15 \text{ and } 2y = 10 \\[1em] \Rightarrow x = 5 \text{ and } y = 5.

Hence, the coordinates of M and N are (5, 0) and (0, 5) respectively.

(ii) Slope of line MN can be given by y2−y1x2−x1\dfrac{y2 - y1}{x2 - x1}

Putting value in above equation we get slope,

=5−00−5=−55=−1.= \dfrac{5 - 0}{0 - 5} \\[1em] = -\dfrac{5}{5} \\[1em] = -1.

Hence, the slope of the line is -1.

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