Question

Rohan and Sohan stay in a village which are 50 km apart and are situated on Delhi Agra Highway as shown in the figure. Another highway YY’ crosses Agra Delhi highway at O (0, 0). A small local road PQ crosses both the highways at points A and B such that OA = 10 km and OB = 12 km. Also the villages of Sita and Gita are on the smaller highway YY’. Sita’s village is 12 km from O and Gita’s village is 15 km from O.

Now answer the following questions:

1. Find the Coordinates of A and B.
2. Find the Length of local road AB.
3. What is the shortest distance Sohan will cover to reach Sita’s village.
4. How far is Rohan’s village from Gita’s village?

Answer 1

(i) Coordinates of Points A and B

• The Delhi-Agra Highway represents the x-axis (X'OX) and highway YY' represents the y-axis intersecting perpendicularly at the origin O(0, 0) [1.1].
• Point A lies on the positive x-axis at a distance of OA = 10 km from origin (O). ∴ Coordinates of A = (10, 0)
• Point B lies on the positive y-axis at a distance of OB = 12 km from origin (O). ∴ Coordinates of B = (0, 12)

(ii) Length of Local Road AB

Given: Points [A(10, 0)] and [B(0, 12)]

Formula: Distance \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

\[\text{AB} = \sqrt{(0 - 10)^2 + (12 - 0)^2}\]

\[\text{AB} = \sqrt{(-10)^2 + 12^2}\]

\[\text{AB} = \sqrt{100 + 144} = \sqrt{244} = 2\sqrt{61}\text{ km} \approx 15.62\text{ km}\]

(iii) Shortest Distance Covered by Sohan to Sita's Village

Sita's village is situated on highway YY' at point B, at a distance of 12 km from (O): Coordinates of Sita's village = (0, 12)

Sohan's village is situated on the Delhi-Agra Highway (x)-axis:

With Rohan and Sohan located 50 km apart symmetrically about origin (O), Sohan's village is at S(25, 0) and Rohan at R(-25, 0):

\[d_{\text{Sohan-Sita}} = \sqrt{(25 - 0)^2 + (0 - 12)^2} = \sqrt{25^2 + (-12)^2} = \sqrt{625 + 144} = \sqrt{769} \approx 27.73\text{ km}\]

(Note: If Sohan's village is marked at x = 35 km, the distance is \(\sqrt{35^2 + 12^2} = \sqrt{1225 + 144} = \sqrt{1369} = 37\text{ km}\); if marked at x = 5 km, the distance is \(\sqrt{5^2 + 12^2} = 13\text{ km})\).

(iv) Distance Between Rohan's Village and Gita's Village

Gita's village is situated on highway YY' at a distance of 15 km from O: Coordinates of Gita's village = (0, -15)

Rohan's village is situated on the negative (x)-axis:

For the symmetric placement R(-25, 0):

\[d_{\text{Rohan-Gita}} = \sqrt{(-25 - 0)^2 + (0 - (-15))^2} = \sqrt{(-25)^2 + 15^2} = \sqrt{625 + 225} = \sqrt{850} = 5\sqrt{34} \approx 29.15\text{ km}\]

(Note: If Rohan's village is marked at R(-20, 0), the distance is \(\sqrt{(-20)^2 + (-15)^2} = \sqrt{400 + 225} = \sqrt{625} = 25\text{ km})\).