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If the circle x2+y2+2gx+2fy+c=0 cuts the three circles x2+y25=0, x2+y28x6y+10=0 and x2+y24x+2y2=0 at the extremities of their diameter, then which of the following is/are true?
(a) c=5
(b) f.g=14725
(c) g+2f=c+2
(d) 4f=3g

Answer
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Hint: In order to solve this problem we need to use the concept of the orthogonal circle, orthogonal circles are those circles which intersect at right angles. We need to compare the coefficients of given circles with the general equation of the circle then use the condition of orthogonal of circles. The formula for condition of orthogonality:
2(g1g2+f1f2)=C1+C2

Complete step-by-step answer:
Orthogonal circles:
When the first circle intersects the three circles at the extremities of the diameter, then they are orthogonal to each other.
Given that the equations of three circles are
C1x2+y25=0, C2x2+y28x6y+10=0 and C3x2+y24x2y2=0
The center of the first circle C1, is at (0,0).
For center of second circle, C2, we have:
2g=8g=82=4
and
2f=6f=62=3
Centre of the second circle, C2, is at (g,f)=(4,3).
Similarly for the center of third circle, C3, we have:
2g=4g=42=2
and
2f=2f=22=1
Centre of the second circle, C3, is at (g,f)=(2,1).
The circle which cuts these circles is given by:
x2+y2+2gx+2fy+c=0
Then its center is:
C4(g,f)
Constant term of circle C1 is 5
Constant term of circle C2 is 10
Constant term of circle C3 is 2
Constant term of the circle which cuts these circle i.e. C4 is c
By using the concept of the orthogonal circle, we’ll get:
For the circle C1andC4
2(g1g4+f1f4)=c1+c42(g(0)+f(0))=c50=c5c=5
Thus, option (a) is incorrect.
Similarly using the concept of the orthogonal for the circle C2andC4, we’ll get
2[g2g4+f2f4]=c2+c4
Rewrite the equation after simplification
2[4(g)+3(f)]=10+c......(1)
Substitute the value of c in equation (1), we’ll get:
2[4g3f]=10+54g3f=1524g+3f=152.....(2)
Use the concept of the orthogonal circle for the circle C3andC4, we’ll get:
2[g3g4+f3f4]=c3+c42[2(g)+(1)(f)]=2+c
Rewrite the equation after simplification:
2g+f=2+522g+f=32
Simplifying the equation, we have:
f=32+2g ....(3)
Substitute the value of the f in the equation (2):
4g+3(32+2g)=1524g+92+6g=152
Simplifying the equation we’ll get:
10g=1529210g=242g=1210
Substitute the value of g back in equation (3):
f=32+2×(1210)f=322410
Simplifying it further, we’ll get:
f=152410f=910
Therefore, the product of f and g is:
fg=910×1210fg=9×610×5fg=2725
Therefore option (b) is incorrect.
Comparing the values of 4f and 3g, we’ll get:
4f=4×(910)4f=3610
and 3g=3×(1210)=3610
So we get:
4f=3g
Hence, option (d) is the correct.
Now, comparing the value of g+2f and c+2, we have:
g+2f=12102×910g+2f=3010=3
Similarly we have:
c+2=5+2=7
So we get:
g+2fc+2
Thus, option (c) is also incorrect.

Hence, only option (d) is correct.

Note: The general equation of circle is given as:
x2+y2+2gx+2fy+c=0
From this general equation, we get the center of the circle as C(g,f) and its radius is r=g2+f2c.
For a two degree equation in x and y to be circle, the coefficients of x2 and y2 must be the same.