Question

A group of class X students goes to a picnic during winter holidays. The positions of three friends Aman, Kirti and Chahat, are shown by the points P, Q and R.

1. Find the distance between P and R.   (1)
2. Is Q the midpoint of PR? Justify by finding the midpoint of PR.   (1)
3. 1. Find the point on the x-axis which is equidistant from P and Q.   (2)
      OR
   2. Let S be a point which divides the line joining PQ in the ratio 2 : 3. Find the coordinates of S.   (2)

Answer 1

i. We need to find distance PR.

PQ = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

Here, x₁ = x coordinate of P = 2

y₁ = y coordinate of P = 5

x₂ = x coordinate of Q = 8

y₂ = y coordinate of Q = 3

Putting values:

PR = `sqrt((8 - 2)^2 + (3 - 5)^2`

= `sqrt((6)^2 + (-2)^2)`

= `sqrt(36 + 4)`

= `sqrt40`

= `sqrt(4 xx 10)`

= `sqrt4 xx sqrt10`

= `2 xx sqrt10`

= `2sqrt(10)`

ii. Midpoint of PR = `((x_1 + x_2)/2, (y_1 + y_2)/2)`

= `((2 + 8)/2,(5 + 3)/2)`

= `(10/2, 8/2)`

= (5, 4)

Thus, Q(4, 4) is not the midpoint of PR.

iii. (A) Let the point on the x-axis be (x, 0).

Since it is equidistant from P and Q:

`sqrt((x - 2)^2 + (0 - 5)^2) = sqrt((x - 4)^2 + (0 - 4)^2)`

Square both sides:

(x − 2)² + 25 = (x − 4)² + 16

x² − 4x + 29 = x² − 8x + 32

4x = 3

x = `3/4`

Therefore the point is `(3/4, 0)`.

OR

(B) Given points P(2, 5) and Q(4, 4).

Let the point be S(x, y), which divides PQ in the ratio 2 : 3.

Finding x:

x = `(m_1x_2 + m_2x_1)/(m_1 + m_2)`

Putting values:

x = `(2 xx 4 + 3 xx 2)/(2 + 3)`

= `(8 + 6)/5`

= `14/5`

Finding y:

y = `(m_1y_2 + m_2y_1)/(m_1 + m_2)`

Putting values:

y = `(2 xx 4 + 3 xx 5)/(2 + 3)`

= `(8 + 15)/5`

= `23/5`

So, the required point is S(x, y) is S`(14/5,23/5)`.