Question

The centre of a circle is (2, 6) and its radius is 13 units. Find x, if P(x, 2x) is a point on the circumference of the circle.

Answer 1

Given:

Centre of the circle: C = (2, 6)

Radius: r = 13r

Point on circumference: P = (x, 2x)

Step 1: Use the Distance Formula

`"Distance" = sqrt((x_2 − x_1)^2 + (y_2 − y_1)^2)`

`13 = sqrt((x − 2)^2 + (2x − 6)^2)`

Step 2: Square both sides

(x − 2)² + (2x − 6)² = 169

Step 3: Expand both expressions

(x − 2)² = x² − 4x + 4

`(2x − 6)^2 = 4x^2 − 24x + 36`

x² − 4x + 4 + 4x² − 24x + 36 = 169

5x² − 28x + 40 = 169

Step 4: Bring all terms to one side

5x² − 28x − 129 = 0

Step 5: Solve using the quadratic formula

x = `(−28) ± sqrt((−28)^2 − 4(5) (−129))/(2(5))`

x = `(28 ± sqrt(784 + 2580))/(10)`

x = `(28 ± sqrt3364)/10`

x = `sqrt3364 = 58`

x = `(28 ± 58)/10`

x = `(28 + 58)/10`

x = `86/10`

x = `43/5`

x = `(28−58)/10`

x = `(−30)/10`

x = −3