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# If Q(0, 1) is equidistant from P(5, - 3) and R (x, 6), find the values of x. Also, find the distances QR and PR

**Solution:**

We know that the distance between the two points is given by the Distance Formula = √[( x₂_{ - }x₁_{ })^{2} + (y₂ - y₁)^{2}]

Q (0, 1) is equidistant from P (5, - 3) and R (x, 6).

So, PQ = QR

Hence by applying the distance formula we get,

√(5 - 0)² + (-3 - 1)² = √(0 - x)² + (1 - 6)²

√(5)² + (- 4)² = √(- x)² + (- 5)²

By squaring both the sides,

25 + 16 = x^{2} + 25

16 = x^{2}

x = ± 4

Therefore, point R is (4, 6) or (- 4, 6).

Case (1): When point R is (4, 6),

Distance between P (5, - 3) and R (4, 6) can be calculated using the Distance Formula as,

PR = √(5 - 4)² + (- 3 - 6)²

= √1² + (- 9)²

= √1 + 81

= √82

Distance between Q (0, 1) and R (4, 6) can be calculated using the distance formula as,

QR = √(0 - 4)² + (1 - 6)²

= √(- 4)² + (- 5)²

= √16 + 25

= √41

Case (2): When point R is (- 4, 6)

Distance between P (5, - 3) and R (- 4, 6) can be calculated using the distance formula as,

PR = √(5 - (- 4))² + (- 3 - 6)²

= √(9)² + (- 9)²

= √81 + 81

= 9√2

Distance between Q (0, 1) and R (- 4, 6) can be calculated using the distance formula as,

QR = √(0 - (- 4))² + (1 - 6)²

= √(4)² + (- 5)²

= √16 + 25

= √41

Thus, we see that using R (- 4, 6) we get PR = QR. Thus, the point is R (- 4, 6). Hence, x = - 4.

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 7

**Video Solution:**

## If Q (0, 1) is equidistant from P (5, - 3) and R (x, 6), find the values of x. Also, find the distances QR and PR

NCERT Class 10 Maths Solutions Chapter 7 Exercise 7.1 Question 9

**Summary:**

The value of x for which Q (0, 1) is equidistant from P (5, - 3) and R (x, 6) is - 4. Also, the distances QR and PR are √41 units.

**☛ Related Questions:**

- In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct.
- Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:(i) (- 1, - 2), (1, 0), (- 1, 2), (- 3, 0) (ii) (- 3, 5), (3, 1), (0, 3), (- 1, - 4) (iii) (4, 5), (7, 6), (4, 3), (1, 2)
- Find the point on the x-axis which is equidistant from (2, - 5) and (- 2, 9).
- Find the values of y for which the distance between the points P (2, - 3) and Q (10, y) is 10 units.

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