If $16 l^{2} + 9 m^{2} = 24 l m + 6 l + 8 m + 1$ and if $S$ be the equation of the circle having $l x + m y + 1 = 0$ is a tangent when the equation of the director circle of $S$ is

x^{2} + y^{2} + 6 x + 8 y = 25

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x^{2} + y^{2} - 6 x + 8 y = 25

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x^{2} + y^{2} - 6 x - 8 y = 25

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x^{2} + y^{2} + 6 x - 8 y = 25

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Solution

The correct option is C $x^{2} + y^{2} - 6 x - 8 y = 25$
$∵ 9 l^{2} + 16 m^{2} + 1 + 24 l m + 6 l + 8 m = 25 (l^{2} + m^{2})$
$\Rightarrow {(3 l + 4 m + 1)}^{2} = 25 (l^{2} + m^{2})$
$\Rightarrow \frac{| 3 l + 4 m + 1 |}{\sqrt{(l^{2} + m^{2})}} = 5$
$∴$ Centre $\equiv (3, 4)$ and radius $\equiv 5$
Equation of the circle is ${(x - 3)}^{2} + {(y - 4)}^{2} = 5^{2} = 25$
$∴$ Equation of the director circle is ${(x - 3)}^{2} + {(y - 4)}^{2} = 50$
or $x^{2} + y^{2} - 6 x - 8 y = 25$

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If 16l2+9m2=24lm+6l+8m+1 and if S be the equation of the circle having lx+my+1=0 is a tangent when the equation of the director circle of S is

The correct option is C x2+y2−6x−8y=25∵9l2+16m2+1+24lm+6l+8m=25(l2+m2)⇒(3l+4m+1)2=25(l2+m2)⇒|3l+4m+1|√(l2+m2)=5∴ Centre ≡(3,4) and radius≡5Equation of the circle is (x−3)2+(y−4)2=52=25∴ Equation of the director circle is (x−3)2+(y−4)2=50or x2+y2−6x−8y=25

If $16 l^{2} + 9 m^{2} = 24 l m + 6 l + 8 m + 1$ and if $S$ be the equation of the circle having $l x + m y + 1 = 0$ is a tangent when the equation of the director circle of $S$ is