A) \[5\text{ }({{x}^{2}}+{{y}^{2}})-3x-8y=200\]
B) \[{{x}^{2}}+{{y}^{2}}-4x-8y=200\]
C) \[5\text{ }({{x}^{2}}+{{y}^{2}})-4x=200\]
D) \[{{x}^{2}}+{{y}^{2}}=40\]
Correct Answer: A
Solution :
Let point \[({{x}_{1}},\ {{y}_{1}})\] on the diameter. \[\Rightarrow 2{{x}_{1}}+3{{y}_{1}}=3\] ?.(i) \[16{{x}_{1}}-{{y}_{1}}=4\] ?.(ii) On solving (i) and (ii), we get centre, \[\Rightarrow {{x}_{1}}=\frac{3}{10},\ {{y}_{1}}=\frac{4}{5}\] \[\therefore \] Equation of circle, \[{{(x-{{x}_{1}})}^{2}}+{{(y-{{y}_{1}})}^{2}}={{r}^{2}}\Rightarrow {{\left( x-\frac{3}{10} \right)}^{2}}+{{\left( y-\frac{4}{5} \right)}^{2}}={{r}^{2}}\] \[\because \] Circle passes through (4, 6). So, \[{{r}^{2}}={{\left( \frac{37}{10} \right)}^{2}}+{{\left( \frac{26}{5} \right)}^{2}}\Rightarrow {{r}^{2}}=\frac{4073}{100}\] \[\therefore \] Required equation of circle is \[{{\left( x-\frac{3}{10} \right)}^{2}}+{{\left( y-\frac{4}{5} \right)}^{2}}=\frac{4073}{100}\] \[\Rightarrow 5({{x}^{2}}+{{y}^{2}})-3x-8y=200\].
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