Question

Assertion :Let $A\left(\overline{a}\right)$ & $B\left(\overline{b}\right)$ be two points in space. Let $P\left(\overline{r}\right)$ be a variable point which moves in space such that $\overline{PA}\cdot \overline{PB}\le 0$ such a variable points traces a three-dimensional figure whose volume is given by $\frac{\pi }{6}\{{\overline{a}}^2+{\overline{b}}^2-2\overline{a}\cdot \overline{b}\}\cdot |\overline{a}-\overline{b}|$ Reason: Diameter of sphere subtends acute angle at any point inside the sphere & its volume is given by $\frac{4}{3}\pi r^3$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Statement 1 : $A\left(\overrightarrow{a}\right)$ & $B\left(\overrightarrow{b}\right)$
$\overrightarrow{PA}\cdot \overrightarrow{PA}\le 0$ then locus of P is sphere having diameter $\left|(\overrightarrow{a}-\overrightarrow{b})\right|$ Volume
$\frac{4}{3}\pi {\left|\frac{\overrightarrow{a}-\overrightarrow{b}}{2}\right|}^3=\frac{\pi }{6}{|\overrightarrow{a}-\overrightarrow{b}|}^2\cdot |\overrightarrow{a}-\overrightarrow{b}|$
$=\frac{\pi }{6}({\overrightarrow{a}}^2+{\overrightarrow{b}}^2-2\overrightarrow{a}\cdot \overrightarrow{b})|\overrightarrow{a}-\overrightarrow{b}|$ Hence true

$\text{Statement II : Diameter of sphere subtend acute angle}$
$\text{at point P when the point P moves outside the sphere having radius r}$