Q) Observe the map of Jaipur city placed on a Cartesian plane. Taking Rambagh Palace as origin, the location of some places are given below:

Point A: (−4, 2) Rajasthan High Court
Point B: (4, −4) Birla Mandir
Point C: (4, 3) Heera Bagh
Point D: (−5, −2) Amar Jawan Jyoti
Based on the above, answer the following questions:
(i). Advocate Rehana stays at Heera Bagh. How much distance she has to cover daily to go to the court and coming back home?
(ii). There is a crossing on X-axis which divides AD in a certain ratio. Find the ratio.
(iii). Is Birla Mandir equidistant from Heera Bagh and Amar Jawan Jyoti? Justify your answer.
(iv). Using section formula, show that points A, O and B are not collinear.

(Q 37- 30/4/2 – CBSE 2026 Question Paper)

Ans:

(i). Distance to & fro to court:

Since, Rehana has to travel from Heera bagh to High court & come back, we need to calculate this distance.

According to the distance formula, distance between 2 points (x1,y1) and (x2,y2)

D =

∵ Points given, C (4,3) for Heera Bagh and A (- 4,2) for High court

∴ Distance between C and A =

= = √65 units

∴ Distance between C and A, to & fro = 2 x √65 = 2√65 = 16.12 units

Therefore, the Rehana needs to cover daily approx. 16.12 units to go to the court and coming back home.

(ii). There is a crossing on X-axis which divides AD in a certain ratio. Find the ratio.

By section formula, if a point divides a line joining (x1, y1) and (x2, y2) in the ratio m : n,

then coordinates of the points are given by: (x,y) =

We know that on X-axis, y = 0

Since the x – axis is crossing line AD, then the intersection point will lie on x-axis as well

Therefore, y -coordinate of intersection point = 0

By section formula, y ordinate value is

∴ = 0

Here, our points are: A (−4, 2) and (−5, −2)

∴ = 0

∴ – 2 m + 2 n = 0

∴ 2 m = 2 n

∴ m = n

∴ m : n = 1 : 1

Therefore, the point on X-axis divides AD in ratio of 1 : 1

(iii) Distance of Birla Mandir from Heera Bagh and Amar Jawan Jyoti:

To check on this, we need to calculate distance of Birla Mandir from Heera Bagh and Amar Jawan Jyoti both.

According to the distance formula, distance between 2 points (x1,y1) and (x2,y2)

D =

Distance of Birla Mandir from Heera Bagh:

Here, Points are, B (4, −4) for Birla Mandir and C (4, 3) for Heera Bagh

∴ DBH =

= √49 = 7 units

Distance of Birla Mandir from Amar Jawan Jyoti:

Here, Points are, B (4, −4) for Birla Mandir and D (−5, −2) for Amar Jawan Jyoti:

∴ DBA =

=

= √85 = 9.2 units

Since, both the distances are not same,

Therefore, the Birla Mandir is NOT equidistant from Heera Bagh and Amar Jawan Jyoti.

(iv). Using section formula, show that points A, O and B are not collinear.

We have the points A (−4, 2), O (0, 0), and B (4, −4).

If they were collinear, O would divide AB in a specific ratio k:1.

Using the section formula for the X-coordinate:

x =

∴ 0 =

∴ 0 =

∴ 4 k – 4 = 0

∴ k = 1

With k = 1, y ordinate should also result in 0.

∴ y =

∴ =

∴ =

∵ k = 1

∴ y =

=

=  = – 1

Since the calculated Y-coordinate (- 1) does not match the actual Y-coordinate of the origin (0),

∴ the points A, O, and B are not collinear.

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