MHT CET · Maths · Three Dimensional Geometry

$∆ A B C$ has vertices at $A \equiv (2, 3, 5), B \equiv (- 1, 3, 2)$ and $C \equiv (λ, 5, μ) .$ If the median through $A$ is equally inclined to the axes, then the values of $λ$ and $μ$ respectively are

A $10, 7$
B $9, 10$
C $7, 9$
D $7, 10$

Verified Solution

Answer & Solution

Correct Answer

(D) $7, 10$

Step-by-step Solution

Detailed explanation

D = (\frac{λ - 1}{2}, \frac{5 + 3}{2}, \frac{μ + 2}{2})

= (\frac{λ - 1}{2}, 4, \frac{μ + 2}{2})

Direction ratios of AD are (

\frac{λ - 1}{2}, - 2, 4 - 3, \frac{μ + 2}{2} - 5

)

i . e .(\frac{λ - 1 - 4}{2}, 1, \frac{μ + 2 - 10}{2}

)

i . e .(\frac{λ - 5}{2}, 1, \frac{μ + 8}{2}

)
Now if we go by options (since the line AD is equally inclined to coordinate axes, its direction ratio are in ratio (

\pm 1 : \pm 1 : \pm 1

)
We get that for

λ = 7, μ = 10

direction ratios of AD are

1,1,1 .

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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∆ABC has vertices at A≡2, 3, 5, B≡(-1, 3, 2) and C≡λ, 5, μ. If the median through A is equally inclined to the axes, then the values of λ and μ respectively are

Step-by-step Solution

$∆ A B C$ has vertices at $A \equiv (2, 3, 5), B \equiv (- 1, 3, 2)$ and $C \equiv (λ, 5, μ) .$ If the median through $A$ is equally inclined to the axes, then the values of $λ$ and $μ$ respectively are