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Statement $1$ : The only circle having radius $\sqrt {10} $ and a diameter along line $2x + y = 5$ is $x^2 + y^2 - 6x +2y = 0$.
Statement $2$ : $2x + y = 5$ is a normal to the circle $x^2 + y^2 -6x+2y = 0$.
Statement $1$ is false; Statement $2$ is true
Statement $1$ is true; Statement $2$ is true, Statement $2$ is a correct explanation for Statement $1$.
Statement $1$ is true; Statement $2$ is false
Statement $1$ is true; Statement $2$ is true; Statement $2$ is not a correct explanation for Statement $1$.
Solution
Circle: ${x^2} + {y^2} – 6x – 2y = 0\,\,\,\,\,\,\,\,\,……\left( i \right)$
Line: $2x + y = 5\,\,\,\,\,\,\,\,\,\,\,…….\left( {ii} \right)$
Center $ = \left( {3, – 1} \right)$
Now, $2 \times 3 – 1 = 5$, hence center lies on the given line. Therefore line passes through the center. The given line is normal to the circle.
Thus statemement-$2$ is true, but statemement-$1$is not true as there are infinite circle accordinh to the given conditions.
