If the distances of the point (1,2,a) from the line along the lines are equal, then a+b+c is [2026]

Question

If the distances of the point (1,2,a) from the line $\frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{1}$ along the lines $L_{1} : \frac{x - 1}{3} = \frac{y - 2}{4} = \frac{z - a}{b} and L_{2} : \frac{x - 1}{1} = \frac{y - 2}{4} = \frac{z - a}{c}$ are equal, then $a + b + c$ is equal to [2026]

Smart Achievers · Accepted Answer

(3)

$L : \frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{1}$

$L_{1} : \frac{x - 1}{3} = \frac{y - 2}{4} = \frac{z - a}{b} = λ$

$L_{2} : \frac{x - 1}{1} = \frac{y - 2}{4} = \frac{z - a}{c} = μ$

$Let A (3 λ + 1, 4 λ + 2, b λ + a)$

$It lies on L$

$∴$ $\frac{3 λ}{1} = \frac{4 λ + 2}{2} = \frac{b λ + a - 1}{1}$

$\Rightarrow λ = 1 and a + b - 1 = 3$

$\Rightarrow A (4, 6, 4), a + b = 4 . . . (1)$

$Let B (μ + 1, 4 μ + 2, c μ + a)$

$It also lies on L$

$\frac{μ}{1} = \frac{4 μ + 2}{2} = \frac{c μ + a - 1}{1}$

$\Rightarrow 2 μ = 4 μ + 2$

$\Rightarrow μ = - 1$

$a - c - 1 = - 1$

$\Rightarrow a = c . . . (2), B (0, - 2, 0)$

$Also P A = P B, P (1, 2, a), A (4, 6, 4)$

$\Rightarrow 9 + 16 + {(a - 4)}^{2} = 1 + 16 + a^{2}$

$\Rightarrow 16 + 8 = 8 a$

$\Rightarrow a = 3$ $∴$ $c = 3, b = 1$

$∴$ $a + b + c = 7$

Solution

Q.
If the distances of the point (1,2,a) from the line $\frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{1}$ along the lines $L_{1} : \frac{x - 1}{3} = \frac{y - 2}{4} = \frac{z - a}{b} and L_{2} : \frac{x - 1}{1} = \frac{y - 2}{4} = \frac{z - a}{c}$ are equal, then $a + b + c$ is equal to [2026]

1 5

2 6

3 7

4 4

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Solution

Q. If the distances of the point (1,2,a) from the line x-11=y2=z-11 along the lines L1: x-13=y-24=z-ab and L2: x-11=y-24=z-ac are equal, then a+b+c is equal to [2026]

1 5

2 6

3 7

4 4

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Q.
If the distances of the point (1,2,a) from the line $\frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{1}$ along the lines $L_{1} : \frac{x - 1}{3} = \frac{y - 2}{4} = \frac{z - a}{b} and L_{2} : \frac{x - 1}{1} = \frac{y - 2}{4} = \frac{z - a}{c}$ are equal, then $a + b + c$ is equal to [2026]