Question

A circle centered at (2, 1) passes through the points A(5, 6) and B(–3, K). Find the value(s) of K. Hence find length of chord AB.

Answer 1

First, calculate the radius squared (OA²):

OA² = (5 – 2)² + (6 – 1)²

OA² = (3)² + (5)²

= 9 + 25

= 34

Now, calculate OB² and set it equal to 34:

OB² = (–3 – 2)² + (K – 1)² = 34

(–5)² + (K – 1)² = 34

25 + (K – 1)² = 34

(K – 1)² = 9

K – 1 = ±3

Case 1: K – 1 = 3

⇒ K = 4

Case 2: K – 1 = –3

⇒ K = –2

Calculating length of chord AB:

If K = 4, points are A(5, 6) and B(–3, 4):

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

`AB = sqrt((-3 - 5)^2 + (4 - 6)^2`

= `sqrt((-8)^2 + (-2)^2`

= `sqrt(64 + 4)`

= `sqrt(68)`

= `2sqrt(17)` units

If K = –2, points are A(5, 6) and B(–3, –2):

`AB = sqrt((-3 - 5)^2 + (-2 - 6)^2`

= `sqrt((-8)^2 + (-8)^2`

= `sqrt(64 + 64)`

= `sqrt(128)`

= `8sqrt(2)` units

The values of K are 4 and –2.

The corresponding lengths of chord AB are `2sqrt(17)` units and `8sqrt(2)` units.