JEE Main & Advanced Mathematics Three Dimensional Geometry Image of a Point in a Plane

Let P and Q be two points and let \[\pi \] be a plane such that

(i) Line PQ is perpendicular to the plane \[\pi ,\] and

(ii) Mid-point of PQ lies on the plane \[\pi \].

Then either of the point is the image of the other in the plane\[\pi \].

To find the image of a point in a given plane, we proceed as follows

(i) Write the equations of the line passing through P and normal to the given plane as  \[\frac{x-{{x}_{1}}}{a}=\frac{y-{{y}_{1}}}{b}=\frac{z-{{z}_{1}}}{c}\].

(ii) Write the co-ordinates of image Q as \[({{x}_{1}}+ar,\,{{y}_{1}},\,+br,\,{{z}_{1}}+cr)\].

(iii) Find the co-ordinates of the mid-point R of PQ.

(iv) Obtain the value of r by putting the co-ordinates of R in the equation of the plane.

(v) Put the value of r in the co-ordinates of Q.