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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 co-ordinates of M and N

(ii) slope of the line MN

Straight Line Eq

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Answer

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 : Equation of a Straight line, RSA Mathematics Solutions ICSE Class 10.

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

⇒ 3x = 15 and 2y = 10

⇒ x = 5 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}

Substituting value in above equation we get slope,

= 5−00−5\dfrac{5 - 0}{0 - 5}

= 5−5\dfrac{5}{-5}

= -1.

Hence, the slope of the line is -1.

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