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Mathematics

If a point R(235,335)R\Big(\dfrac{23}{5}, \dfrac{33}{5}\Big) divides the line segment PQ joining the points P(3, 5) and Q(x, y) in the ratio 2 : 3 internally, then the values of x and y respectively are :

  1. 4, 7

  2. 5, 9

  3. 7, 8

  4. 7, 9

Section Formula

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Answer

Given,

Point R = (235,335)\Big(\dfrac{23}{5}, \dfrac{33}{5}\Big) and P(3, 5), Q(x, y).

divides the line segment PQ joining the points P(3, 5) and Q(x, y) in the ratio 2 : 3 internally, then the values of x and y respectively are :Reflection, RSA Mathematics Solutions ICSE Class 10.

Given,

m1 : m2 = 2 : 3

By section-formula,

(x, y) = (m1x2+m2x1m1+m2,m1y2+m2y1m1+m2)\Big(\dfrac{m1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big)

Substituting values we get :

⇒(235,335)=(2x+3×32+3,2y+3×52+3)⇒235=2x+95,335=2y+155⇒23=2x+9,33=2y+15⇒23−9=2x,33−15=2y⇒14=2x,18=2y⇒x=142,y=182⇒x=7,y=9.\Rightarrow \Big(\dfrac{23}{5}, \dfrac{33}{5}\Big) = \Big(\dfrac{2x + 3 \times 3}{2 + 3}, \dfrac{2y + 3 \times 5}{2 + 3}\Big) \\[1em] \Rightarrow \dfrac{23}{5} = \dfrac{2x + 9}{5}, \dfrac{33}{5} = \dfrac{2y + 15}{5} \\[1em] \Rightarrow 23 = 2x + 9, 33 = 2y + 15 \\[1em] \Rightarrow 23 - 9 = 2x, 33 - 15 = 2y \\[1em] \Rightarrow 14 = 2x, 18 = 2y \\[1em] \Rightarrow x = \dfrac{14}{2}, y = \dfrac{18}{2} \\[1em] \Rightarrow x = 7, y = 9.

Therefore, the values of x and y are 7 and 9 respectively.

Hence, Option 4 is the correct option.

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