A) \[{{x}^{2}}+{{y}^{2}}-2x+3y-3=0\]
B) \[{{x}^{2}}+{{y}^{2}}+2x-3y-5=0\]
C) \[2{{x}^{2}}+2{{y}^{2}}-2x+5y-8=0\]
D) None of these
Correct Answer: A
Solution :
Let the equation be \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] ?.(i) But it passes through \[(-1,\ -3)\] and (3, 0) therefore \[10-2g-6f+c=0\] ?.(ii) \[9+6g+c=0\] ?.(iii) Also centre is \[C(-g,\ -f)\]. Slope of tangents \[=-\frac{4}{3}\Rightarrow \] Slope of normal \[=\frac{3}{4}\] \[\Rightarrow \frac{f}{3+g}=\frac{3}{4}\Rightarrow 3g-4f+9=0\] .....(iv) Now on solving (ii), (iii) and (iv), we get \[g=-1,\ f=\frac{3}{2}\] and \[c=-3\] Therefore, the equation of circle is \[{{x}^{2}}+{{y}^{2}}-2x+3y-3=0\]. Trick : The points (?1, ?3) and (3, 0) must satisfy the equation of circle. Circle given in (a) satisfies both the points. Also check whether it touches the line \[4x+3y-12=0\] or not.
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