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Consider the following for the next two (02) items that follow.
A circle is passing through the points
(5,−8),
(−2,9) and
(2,1).
[NDA-II 2021]
(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=0â‹…â‹…â‹…â‹…â‹…â‹…â‹…(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=0â‹…â‹…â‹…â‹…â‹…â‹…â‹…(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=24From Eq. (vi)
g=2f+10 g=58From Eq.
(iv)
c=−4g−2f−5 =−232−48−5 c=−285 ∴ Centre =(−g,−f) =(−58,−24)© examsnet.com