Manipal Entrance Test 2015 Math Paper

Let the equation of the circle be x2+y2+2gx+2fy+c=0
which passes through (0, 0) and (1, 0).
∴c=0 and 1+2g+c=0
⇒c=0,g=−

1

2

The circle x2+y2+2gx+2fy+c=0 touches the circle x2+y2=9
∴√g2+f2=3±√g2+f2−c [∵C1C2=r1±r2]
⇒√

1

4

+f2=3±√

1

4

+f2
⇒3=2√

1

4

+f2
⇒f2=

9

4

−

1

4

=2
⇒f=±√2
Hence, the coordinates of the centre are (

1

2

,±√2)