Question

The centre of a circle is (x – 2, x + 1) and it passes through the points (4, 4). Find the value (or values) of x, if the diameter of the circle is of length 2 5 units .

(a) 1 or 3 

(b) –1 or 4 

(c) 5 or 4 

(d) 3 or –2

Answer 1

(c) 5 or 4

Radius of the circle = Dist. between the centre and given pt. on the circle

= \(\sqrt{(x-2-4)^2+(x+1-4)^2}\)

= \(\sqrt{(x-6)^2+(x-3)^2}\)

= \(\sqrt{x^2-12x+36+x^2-6x+9}\)

= \(\sqrt{2x^2-18x+45}\)

Given, diameter = 2√5

⇒ radius = \(\frac12\times2\sqrt5 = \sqrt5\)

∴ \(\sqrt{2x^2-18x+45}\) = √5

⇒ 2x² – 18x + 45 = 5 

⇒ 2x² – 18x + 40 = 0 

⇒ x² – 9x + 20 = 0 

⇒ (x – 5) (x – 4) = 0 

⇒ x = 5 or 4