Question

Find the equation of a circle that has its centre on
the line x = 1 - y, passing through origin & (4,2).
help me​

Answer 1

Step-by-step explanation:

ANSWER

Equation of circle which passes through the origin is x

2

+y

2

+2gx+2fy+c=0 .....(1)

the centre of the circle (1) is (−g,−f)

if the centre lies on the line x+y=4

then −g−f=4⇒g+f=−4 .....(2)

the given equation of the orthogonal circle is x

2

+y

2

−4x+2y+4=0 ....(3)

Comparing the circle (2) with the general equation of the circle, we get

g

1

=−2;f

1

=1 and c

1

=4

the circle (1) is orthogonal to circle (2)

∴2gg

1

+2ff

1

=c+c

1

if two circles x

2

+y

2

+2g

1

x+2f

1

y+c

1

=0 and x

2

+y

2

+2g

2

x+2f

2

y+c

2

=0 are orthogonal then 2g

1

g

2

+2f

1

f

2

=c

1

+c

2

⇒2g(−2)+2f(1)=0+4

⇒−4g+2f=4 or −2g+f=2 .....(4)

Solving eqn(2) and eqn(4) we get

g+2g=−4−2 or 3g=−6 or g=−2

f=−4−g=−4−(−2)=−4+2=−2 by subustituting for g=−2

Thus, the equation of the required circle is x

2

+y

2

+2×−2x+2×−2y=0

or x

2

+y

2

−4x−4y=0