The equation of the circle C is 2x 2 + 2y 2 − 6x − 10y − 15 = 0. Which of the following is/are true?

Question

I. The coordinates of the centre of C are (3, 5).
II. The origin lies inside C.
III. The area of C is smaller than 55 square units.

Answer 1

General form of the circle : 2x^2 + 2y^2 -6x -10y -15 =0

So, standard form of the circle is

(x - 1.5)^2 + (y-2.5)^2 = 4^2

Therefore center lies in (3/2 , 5/2), So (I) is false.

Distance from center of circle to origin = sqrt( (1.5-0)^2  + (2.5-0)^2 ) = sqrt(8.5)

Since distance from center of circle to origin, sqrt(8.5) < radius of circle, 4

Therefore option (II) is true

Area of circle = pi * 4 * 4= 3.14 * 16 = 50.24 sq unit (approx)

Since 50.24  < 55, therefore option (III) is true