Animated Solution for Mathematics - Three Dimensional Geometry: Let $a,b\in \mathbb{R}$. If the mirror image of the point $P(a,6,9)$ with respect to the line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is $(20,b,-a-9)$, then $|a+b|$ is equal to:

In the realm of 3D geometry, reflecting a point $P(a,6,9)$ across a line $L$ defined by results in an image point $Q(20,b,-a-9)$. This process relies on the principle of symmetry, where the line $L$ acts as the perpendicular bisector of the segment $PQ$.

Our Anchor in the Void

The midpoint $M$ of the segment $PQ$ must lie on the line $L$. We calculate the coordinates of $M$ by averaging the coordinates of $P$ and $Q$:

Simplifying the components, we obtain:

Since $M$ lies on the line $L$, it must satisfy the given symmetric equations of the line.

Substituting the coordinates of $M$ into the line equation, we get:

Simplifying the numerators, the expression becomes:

To determine the value of $a$, we equate the first and third ratios:

Cross-multiplying and simplifying, we have $18(a+14)=14(a+2)$, which reduces to $9(a+14)=7(a+2)$. Expanding this yields $9a+126=7a+14$, leading to $2a=-112$, or $a=-56$.

Next, we solve for $b$ by equating the first and second ratios:

This results in $b+2=-30$, which gives $b=-32$.

With the values of $a$ and $b$ determined, we calculate the final required value:

Through the application of geometric symmetry and algebraic substitution, we have successfully resolved the coordinates and reached the final result of 88.