Question

$P$ is any point on the plane $lx+my+nz=p$ a point $Q$ is taken on the line $OP$ such that $OP.OQ=p^2$, then the locus of $Q$ is

• A. $lx+my+nz=p(x^2+y^2+z^2)$
• B. $p(lx+my+nz)=(x^2+y^2+z^2)$
• C. $p(x+y+z)=l{x}^2+m{y}^2+n{z}^2$
• D. None of these

Answer 1

Answer: (B)

$p(lx+my+nz)=(x^2+y^2+z^2)$

Let $Q$ be the point $(x^{\prime },y^{\prime },z^{\prime })$.

Also, let $OP=r$ and $OQ=r^{\prime }=\sqrt{({x^{\prime }}^2+{y^{\prime }}^2+{z^{\prime }}^2)}$.

Then, equation of $OQ$(in distance form) is

$\frac{x}{\frac{x^{\prime }}{r^{\prime }}}=\frac{y}{\frac{y^{\prime }}{r^{\prime }}}=\frac{z}{\frac{z^{\prime }}{r^{\prime }}}=r$

So, coordinate of the point$P$, which is at a distance $r$ from $O$ are

$(r\frac{x^{\prime }}{r^{\prime }},r\frac{y^{\prime }}{r^{\prime }},r\frac{z^{\prime }}{r^{\prime }})\Rightarrow (\frac{x^{\prime }{p}^2}{{r^{\prime }}^2},\frac{y^{\prime }{p}^2}{{r^{\prime }}^2},\frac{z^{\prime }{p}^2}{{r^{\prime }}^2})$ as $r{r}^{\prime }=p^2$

This point $P$ lies on the plane $lx+my+nz=p$

$\therefore l\frac{x^{\prime }{p}^2}{{r^{\prime }}^2}+m\frac{y^{\prime }{p}^2}{{r^{\prime }}^2}+n\frac{z^{\prime }{p}^2}{{r^{\prime }}^2}=p\Rightarrow p(l{x}^{\prime }+m{y}^{\prime }+n{z}^{\prime })={r^{\prime }}^2=({x^{\prime }}^2+{y^{\prime }}^2+{z^{\prime }}^2)$

$\therefore$ Locus of $Q(x^{\prime },y^{\prime }{z}^{\prime })$ is

$p(lx+my+nz)=(x^2+y^2+z^2)$