Image of point P (1, 2, a) with respect to line mirror \(\frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}\) is point Q (5, b, c), then value of \((a^2 + b^2 + c^2)\) is :
Show Hint
Finding the image of a point in a line involves two key geometric conditions: the midpoint of the point and its image lies on the line, and the line connecting the point and its image is perpendicular to the mirror line.
Step 1: Problem Understanding:
We are given a point \( P(1, 2, a) \), a line with equation \( \frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2} \), and its image point \( Q(5, 8, c) \). Our goal is to find the values of \(a\), \(b\), and \(c\) using the given geometric properties and calculate the sum \( a^2 + b^2 + c^2 \).
Step 2: Geometrical Properties: Property 1: Midpoint Condition
The midpoint of the segment joining \(P\) and \(Q\) lies on the line of reflection. We calculate the midpoint \( M \) of points \( P(1, 2, a) \) and \( Q(5, 8, c) \), given by:
\[
M = \left( \frac{1+5}{2}, \frac{2+8}{2}, \frac{a+c}{2} \right) = (3, 5, \frac{a+c}{2}).
\]
Since this midpoint lies on the line, the coordinates of \( M \) must satisfy the line’s equation:
\[
\frac{3-6}{3} = \frac{5-7}{2} = \frac{\frac{a+c}{2}-7}{-2}.
\]
This simplifies to:
\[
-1 = \frac{b-12}{4} = \frac{a+c-14}{-4}.
\]
Solving these, we find \( b = 8 \) and \( a + c = 18 \).
Property 2: Perpendicularity Condition
The segment \( PQ \) is perpendicular to the line. The direction ratios of the line segment \( PQ \) are \( (4, b-2, c-a) \), which simplify to \( (4, 6, c-a) \) when \( b = 8 \). The direction ratios of the line are \( (3, 2, -2) \).
The condition for perpendicularity is:
\[
4 \times 3 + 6 \times 2 + (c-a) \times (-2) = 0.
\]
This simplifies to:
\[
24 = 2(c-a), \quad \text{which gives} \quad c - a = 12.
\]
Step 3: Solving for \( a \), \( b \), and \( c \)
We now solve the system of equations:
\( a + c = 18 \),
\( c - a = 12 \).
By adding these equations, we get \( 2c = 30 \), hence \( c = 15 \). Substituting \( c = 15 \) into \( a + c = 18 \), we get \( a = 3 \).
Step 4: Final Calculation
Now, we calculate \( a^2 + b^2 + c^2 \):
\[
a^2 + b^2 + c^2 = 3^2 + 8^2 + 15^2 = 9 + 64 + 225 = 298.
\]
Thus, the final value is 298.
