Let the circle S : 36 x 2 + 36 y 2 - 108 x + 120 y + C = 0 be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, x - 2 y = 4 and 2 x - y = 5 lies inside the circle S , then:

Step-by-step solution

The given circle is S : 36 x 2 + 36 y 2 - 108 x + 120 y + C = 0 ⇒ x 2 + y 2 - 3 x + 10 3 y + C 36 = 0 We know that the centre and radius of a circle x 2 + y 2 + 2 g x + 2 f y + c = 0 is - g ,   - f and g 2 + f 2 - c respectively. Thus, centre e ≡ - g , - f ≡ 3 2 ,   - 10 6 and radius = r = 9 4 + 100 36 - C 36 Now, this circle neither intersects nor touches the co-ordinate axes, hence, the radius of the circle is less than the absolute value of the co-ordinate with smaller value. ⇒