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.

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 = x² + 25

16 = x²

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.