Question

Let $A(0,6,8),B(15,20,0)$ are two given points and $P(\lambda ,0,0)$ is a point on $x$ - axis such that $PA+PB$ is minimum. Then which of the following is/are correct?

• A. Perpendicular distance of the origin from the plane passing through $P,A,B$ is $\frac{10}{3}$
• B. Perpendicular distance of the origin from the plane passing through $P,A,B$ is $\frac{10}{9}$
• C. Value of $\lambda$ is $5$
• D. Value of $\lambda$ is $10$

Answer 1

Answer: (A), (C)

The correct options are
A Perpendicular distance of the origin from the plane passing through $P,A,B$ is $\frac{10}{3}$
C Value of $\lambda$ is $5$

$PA+PB=\sqrt{{\lambda }^2+100}+\sqrt{(\lambda -15{)}^2+400}$is same as finding minimum value of $P^{\prime }{A}^{\prime }+P^{\prime }{B}^{\prime }$ where $P^{\prime }(\lambda ,0),A^{\prime }(0,10)$ and $B^{\prime }(15,-20)$ which is possible only when $P^{\prime },A^{\prime },B^{\prime }$ are collinear
$\Rightarrow \lambda =5$
Equation of plane passing through $P(5,0,0),A(0,6,8)\&B(15,20,0)$ is
$2x-y+2z=10$
So distance $=\frac{10}{3}$