Question

If the image of the point $P(1,-2,3)$ in the plane, $2x+3y-4z+22=0$

measured parallel to the line, $\frac{x}{1}=\frac{y}{4}=\frac{z}{5}$ is $Q$, the $PQ$ is equal to.

• A. $3\sqrt{5}$
• B. $2\sqrt{42}$
• C. $\sqrt{42}$
• D. $6\sqrt{5}$

Answer 1

The correct option is B $2\sqrt{42}$

Equation of plane: $P$

$2x+3y-4z+22=0$

Direction ratio of plane are: $(2,3,-4)$

Equation of line: $L_1$

$\frac{x}{1}=\frac{y}{4}=\frac{z}{5}$

Direction ratio of line are: $(1,4,5)$

Equation of a line $PQ$ passing through point $P(1,-2,3)$ along line $L_1$ is:

$\frac{x-1}{1}=\frac{y+2}{4}=\frac{z-3}{5}=k$

$\Rightarrow x=k+1,y=4k-2,z=5k+3$

Lets say $R$ is the mid point of the line $PQ$, therefore $R$ lie on the plane.

Hence, it should satisfy the equation of the plane.

$2(k+1)+3(4k-2)-4(5k+3)+22=0$

$\Rightarrow k=1$

Coordinates of $R=(2,2,8)$

Length of the image $PQ=2PR$

$\Rightarrow PQ=2\sqrt{(2-1{)}^2+(2+2{)}^2+(8-3{)}^2}=2\sqrt{42}$