Statement 1 : The only circle having radius&n | vedclass

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Question

Circle: ${x^2} + {y^2} - 6x - 2y = 0\,\,\,\,\,\,\,\,\,......\left( i \right)$

Line: $2x + y = 5\,\,\,\,\,\,\,\,\,\,\,.......\left( {ii} \

Vedclass · Accepted Answer

Statement $1$ is false; Statement $2$ is true

Vedclass · Answer

Circle: ${x^2} + {y^2} - 6x - 2y = 0\,\,\,\,\,\,\,\,\,......\left( i ight)$

Line: $2x + y = 5\,\,\,\,\,\,\,\,\,\,\,.......\left( {ii} ight)$

Center $ = \left( {3, - 1} ight)$

Now, $2 	imes 3 - 1 = 5$, 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.

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 $x^2 + y^2 -6x+2y = 0$.

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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 $x^2 + y^2 -6x+2y = 0$.