JEE Advanced 2018 · Paper 2 · Mathematics Q12

← JEE Advanced 2018 · Paper 2

MathematicsNumerical+3Question 48 of 54

Let $P$ be a point in the first octant, whose image $Q$ in the plane $x + y = 3$ (that is, the line segment $PQ$ is perpendicular to the plane $x + y = 3$ and the mid-point of $PQ$ lies in the plane $x + y = 3$ ) lies on the $z$ -axis. Let the distance of $P$ from the $x$ -axis be 5. If $R$ is the image of $P$ in the $xy$ -plane, then the length of $PR$ is _____ .

Your answer

Answer and worked solution

Official answer: 8

Set up. Let $P=(a,b,c)$ . Its image $Q$ is on the $z$ -axis, and $PQ$ is perpendicular to $x+y=3$ (normal $(1,1,0)$ ). So $Q=(0,0,c)$ and $(a,b,0)\parallel(1,1,0)$ , giving $a=b$ .

Mid-point. $\left(\tfrac a2,\tfrac b2,c\right)$ lies on the plane: $a+b=6$ , so $a=b=3$ .

Distance from the $x$ -axis. $\sqrt{b^2+c^2}=5$ , so $c=4$ .

Length. $R=(3,3,-4)$ , so $PR=2c=8$ .

Answer: 8

Topics:Angles and distances Equation of a plane

← Previous question Next question →

JEE Advanced 2018 · Paper 2, Mathematics question 12