.. The required circle is x + y - 41
Ex 2.1.6 :
x2 + y2 - 4x + 2y – 7 = 0 and having the centre at (2, 3).
areh
Find the equation of the circle which cuts orthogonally the
Answers
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
hope it helped
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Answer:
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