Question

$P$ is any point on the plane $lx+my+nz=p$. A point $Q$ taken on the line $OP($where $O$ is the origin$)$ such that $OP.OQ=p^2$, then the locus $Q$ is

• A. $p(lx+my+nz)=x^2+y^2+z^2$
• B. $p^2(lx+my+nz)=x^3+y^3+z^3$
• C. $(x^2+y^2+z^2)(lx+my+nz)=p(x^2+y^2+z^2)$
• D. None of these

Answer 1

The correct option is A $p(lx+my+nz)=x^2+y^2+z^2$

Let $P=(x,y,z)$ which lies on the plane $lx+my+nz=p.$

Also since $Q$ lies on line $OP.OQ=(ax,ay,az)$ where $a$ is a constant.

Now $OP.OQ=a(x^2+y^2+z^2)=p^2=p(lx+my+nz)$

Now let $ax=X,ay=Y$ and $az=Z.$

Thus substituting these, we get $(X^2+Y^2+Z^2)=p(lX+mY+nZ)$