JEE Advanced · Mathematics · 31. 3D Geometry

Let $α, β, γ, δ$ be real numbers such that $α^{2} + β^{2} + γ^{2} \neq 0$ and $α + γ = 1$ . Suppose the point $(3, 2, - 1)$ is the mirror image of the point $(1, 0, - 1)$ with respect to the plane $α x + β y + γ z = δ$ . Then which of the following statements is/are TRUE?

A $α + β = 2$
B $δ - γ = 3$
C $δ + β = 4$
D $α + β + γ = δ$

Verified Solution

Answer & Solution

Correct Answer

(A) $α + β = 2$

Step-by-step Solution

Detailed explanation

From the diagram, Mid point of

A & A'

lies on the plane, i.e.,

B (2, 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.

Same subject

Explore more questions on app

From JEE Advanced

Let

f (x) = x + \log_{e} x - x \log_{e} x, x \in (0, \infty) .

•

Column 1 contains information about zeros of

f (x), f^{'} (x)

and

f^{''} (x) .

•

Column 2 contains information about the limiting behaviour of

f (x), f^{'} (x)

and

f^{''} (x)

at infinity.

•

Column 3 contains information about increasing-decreasing nature of

f (x)

and

f^{'} (x) .

Column 1	Column 2	Column 3
(I) $f (x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f (x) = 0$	(P) $f$ is increasing in (0, 1)
(II) $f^{'} (x) = 0$ for some $x i n (1, e)$	(ii) $\lim_{x \to \infty} f (x) = - \infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f^{'} (x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f^{'} (x) = - \infty$	(R) $f^{'}$ is increasing in (0, 1)
(IV) $f^{''} (x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f^{''} (x) = 0$	(S) $f^{'}$ is decreasing in $(e, e^{2})$

Which of the following options is the only Correct combination?

JEE Advanced 2017 Hard

Explore more questions on app

Let α, β, γ, δ be real numbers such that α2+β2+γ2≠0 and α+γ=1. Suppose the point 3,2,-1 is the mirror image of the point 1,0,-1 with respect to the plane αx+βy+γz=δ. Then which of the following statements is/are TRUE?

Step-by-step Solution

Let $α, β, γ, δ$ be real numbers such that $α^{2} + β^{2} + γ^{2} \neq 0$ and $α + γ = 1$ . Suppose the point $(3, 2, - 1)$ is the mirror image of the point $(1, 0, - 1)$ with respect to the plane $α x + β y + γ z = δ$ . Then which of the following statements is/are TRUE?