Animated Solution for Mathematics - Three Dimensional Geometry: If the mirror image of the point $(1,3,5)$ with respect to the plane $4x-5y+2z=8$ is $(\alpha ,\beta ,\gamma )$, then $5(\alpha +\beta +\gamma )$ equals:

To find the image of the point $P(1,3,5)$ with respect to the plane $4x-5y+2z=8$, we first express the plane in the standard form:

The line connecting the point $P$ and its image $P^{\prime }(\alpha ,\beta ,\gamma )$ must be perpendicular to the plane. This implies the direction ratios of the line are proportional to the normal vector of the plane, which is $(4,-5,2)$.

We utilize the mirror image formula to determine the coordinates of the reflected point:

Here, $(x_1,y_1,z_1)=(1,3,5)$ and $(a,b,c)=(4,-5,2)$. The constant $d$ is $-8$.

First, we evaluate the numerator of the displacement term by substituting the point into the plane equation:

Next, we calculate the denominator, which is the sum of the squares of the normal vector components:

We define the ratio $k$ as follows:

Using the ratio $k=\frac{2}{5}$, we solve for the coordinates $(\alpha ,\beta ,\gamma )$ individually:

For $\alpha$:

For $\beta$:

For $\gamma$:

The sum of the coordinates is:

The problem asks for the value of $5(\alpha +\beta +\gamma )$:

The final result is 47.