Question

$A$: Radius of the sphere $(\overrightarrow{r}-3\hat{i}-4\hat{i}+5\hat{k}).(\overrightarrow{r}-2\hat{i}-3\hat{i}-5\hat{k})=0$ is $\frac{\sqrt{131}}{2}$.

$R$: If $A,B$ are the ends of a diameter of a sphere and $P$ is a point on the sphere, then $\overrightarrow{P}A.\overrightarrow{P}B=0$.

• A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A$
• B. Both $A$ and $R$ are true and $R$ is not correct explanation of $A$
• C. $A$ is true but $R$ is false
• D. $A$ is false but $R$ is true

Answer 1

The correct option is B Both $A$ and $R$ are true and $R$ is not correct explanation of $A$

The equation of a sphere can be written as $(\overline{r}-x).(\overline{r}-y)=0$ where $x$ and $y$ are the ends of a diameter.

Thus, if compared with the assertion, we conclude that $A$ and $B$ are end points of a diameter.
$\Rightarrow \overline{AB}=\overline{i}+7\overline{j}-9\overline{k}$

Thus mod $=\sqrt{1+49+81}=\sqrt{131}$

Hence radius $=\frac{\sqrt{131}}{2}$

The assertion and reason both are correct but the reason doesn't explain why the assertion is correct and so option $B$ is right.