Find the equation of a circle that passes through the point (1, 2) whi

1.	Find the equation of a circle that passes through the point (1, 2) which bisects the circumference of the circle x2 + y2 = 9 and is orthogonal to the circle x2 - y2 − 2x + 8y − 7 = 0.
Answer» Let S' ≡ x² + y² - 9 = 0 S'' ≡ x² + y² - 2x + 8y - 7 = 0 Let S ≡ x² + y² + 2gx + 2fy + c = 0 be the required circle. Since S = 0 passes through (1, 2), we have 2g + 3f + c = -5 ....(1) S = 0 bisects the circumference of S' = 0 S − S' = 0 passes through (0, 0). This gives c + 9 = 0 c = -9 .....(1) S = 0 and S'' = 0 cut orthogonally which implies that 2g(-1) + 2f(4) = c - 7 2g + 8f = c - 7 ....(2) From Eqs. (1)–(2), we have herefore, g = 4,f = -1, and . c =9 Hence the required circle is Let S ≡ x² + y² + 8x - 2y - 9 = 0

Find the equation of a circle that passes through the point (1, 2) which bisects the circumference of the circle x2 + y2 = 9 and is orthogonal to the circle x2 - y2 − 2x + 8y − 7 = 0.

Answer»

Let

S' ≡ x² + y² - 9 = 0

S'' ≡ x² + y² - 2x + 8y - 7 = 0

Let S ≡ x² + y² + 2gx + 2fy + c = 0 be the required circle. Since S = 0 passes through (1, 2), we have

2g + 3f + c = -5 ....(1)

S = 0 bisects the circumference of S' = 0

S − S' = 0 passes through (0, 0). This gives

c + 9 = 0

c = -9 .....(1)

S = 0 and S'' = 0 cut orthogonally which implies that

2g(-1) + 2f(4) = c - 7

2g + 8f = c - 7 ....(2)

From Eqs. (1)–(2), we have

herefore, g = 4,f = -1, and . c =9 Hence the required circle is

Let S ≡ x² + y² + 8x - 2y - 9 = 0

Discussion

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