If the image of the point P(a,2,a) in the line x 2 = y + a 1 = z 1 is Q and the image of Q in the line x - 2 b 2 = y - a 1 = z + 2 b - 5 is P, then a+b is equal to ______. [2026]

Question

If the image of the point $P (a, 2, a)$ in the line $\frac{x}{2} = \frac{y + a}{1} = \frac{z}{1}$ is Q and the image of Q in the line $\frac{x - 2 b}{2} = \frac{y - a}{1} = \frac{z + 2 b}{- 5}$ is P, then $a + b$ is equal to ______. [2026]

Smart Achievers · Accepted Answer

(3)

$\frac{2 λ - a + a}{2} = - 5 μ - 2 b$

$and \frac{a + 4 λ - a}{2} = 2 μ + 2 b$

$and \frac{2 λ - 2 a - 2 + 2}{2} = μ + a$

$\Rightarrow λ - μ = b$

$λ - μ = 2 a$

$λ + 5 μ = - 2 b$

$\Rightarrow b = 2 a and λ = a, μ = - a$

$(a \hat{i} - 2 \hat{j} + 0 \hat{k}) \cdot (2 \hat{i} + \hat{j} + \hat{k}) = 0$

$2 a - 2 = 0$

$a = 1$

$b = 2 a = 2 \times 1 = 2$

$a + b = 1 + 2 = 3$

Solution

Q.
If the image of the point $P (a, 2, a)$ in the line $\frac{x}{2} = \frac{y + a}{1} = \frac{z}{1}$ is Q and the image of Q in the line $\frac{x - 2 b}{2} = \frac{y - a}{1} = \frac{z + 2 b}{- 5}$ is P, then $a + b$ is equal to ______. [2026]

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Solution

Q. If the image of the point P(a,2,a) in the line x2=y+a1=z1 is Q and the image of Q in the line x-2b2=y-a1=z+2b-5 is P, then a+b is equal to ______. [2026]

Ans. (3) 2λ-a+a2=-5μ-2b and a+4λ-a2=2μ+2b and 2λ-2a-2+22=μ+a ⇒λ-μ=b λ-μ=2a λ+5μ=-2b ⇒b=2a and λ=a, μ=-a (ai^-2j^+0k^)·(2i^+j^+k^)=0 2a-2=0 a=1 b=2a=2×1=2 a+b=1+2=3

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Q.
If the image of the point $P (a, 2, a)$ in the line $\frac{x}{2} = \frac{y + a}{1} = \frac{z}{1}$ is Q and the image of Q in the line $\frac{x - 2 b}{2} = \frac{y - a}{1} = \frac{z + 2 b}{- 5}$ is P, then $a + b$ is equal to ______. [2026]