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JEE MainMathematicsModerateCircles

If a circle passes through the point (1,2) and cuts the
circlex2+y2=4 x^{2} + y^{2} = 4\ orthogonally is S=0 and locus of centre of
circle S=0 is lx+my+n=0 S = 0\ is\ \mathcal{l}x + my + n = 0\ then minimum value of
∣l+m+n∣|\mathcal{l +}m + n| is
 (where l,m,n∈I and HCF(l,m,n)=1)\ (where\ \mathcal{l,}m,n \in I\ and\ HCF(\mathcal{l,}m,n) = 1)

Show answer and solution
 Let circle is x2+y2+2gx+2fy+c=0\ Let\ circle\ is\ x^{2} + y^{2} + 2gx + 2fy + c = 0......(i)
This is orthogonal with given circle ⇒c=4\Rightarrow c = 4
circle passes through (1,2)
⇒1+4+2g+4f+4=0\Rightarrow 1 + 4 + 2g + 4f + 4 = 0
⇒2g+4f+9=0\Rightarrow 2g + 4f + 9 = 0
Locus of (−g,−f) is 2x+4y=9Locus\ of\ ( - g, - f)\ is\ 2x + 4y = 9
∣l+m+n∣=∣−3∣=3|\mathcal{l +}m + n| = | - 3| = 3

Source: Grass-G / M-28 Mathematics Paper FST 16 Jan.docx

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