NDA Math Circle

(d) Given that, circle is passing through the points (5,−8),(−2,9) and (2,1) .


Let the equation of circle be
x2+y2+2gx+2fy+c=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(i)
If Eq. (i) passes through (5,−8)
∴‌‌25+64+10g−16f+c=0
⇒‌‌10g−16f+c+89=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(ii)
If Eq. (i) passes through (−2,9)
4+81−4g+18f+c‌‌=0
−4g+18f+c+85‌‌=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(iii)
If Eq. (i) passes through (2,1)
r4+1+4g+2f+c=0
4g+2f+c+5=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(iv)
On solving Eqs. (ii), (iii) and (iv)
Eqs. (ii) - Eq. (iii)
⇒14g−34f+4=0

⇒7g−17f+2=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(v)
Eq. (iv) - Eq. (iii)
⇒‌‌8g−16f−80‌‌=0
g−2f−10‌‌=0‌‌‌⋅⋅⋅⋅⋅⋅⋅(vi)
Eq. (v) −7× Eq. (vi)
−3f+72‌‌=0
f‌‌=24
From Eq. (vi)
g=2f+10
g=58
From Eq.
(iv) c=−4g−2f−5
‌‌=−232−48−5
c‌‌=−285 ∴‌‌‌ Centre ‌‌‌=(−g,−f)
‌‌=(−58,−24)