Let P be the point (3, 4, 12). Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Distance of P from | Value | ||
| (A) | the x-axis | (I) | 3 |
| (B) | the y-axis | (II) | 5 |
| (C) | the z-axis | (III) | \(4\sqrt{10}\) |
| (D) | the line \(\vec{r} = 4\hat{\jmath}+\lambda\hat{k}\) | (IV) | \(3\sqrt{17}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
The foot of the perpendicular from P(x, y, z) to the x-axis is (x, 0, 0), so the distance from the x-axis uses only the other two coordinates: \(\sqrt{y^2+z^2}\). The same holds for the other axes.
(A) the x-axis
This matches (III) \(4\sqrt{10}\):
\(\sqrt{4^2+12^2} = \sqrt{160} = 4\sqrt{10}\)
(B) the y-axis
This matches (IV) \(3\sqrt{17}\):
\(\sqrt{3^2+12^2} = \sqrt{153} = 3\sqrt{17}\)
(C) the z-axis
This matches (II) 5:
\(\sqrt{3^2+4^2} = 5\)
(D) the line \(\vec{r} = 4\hat{\jmath}+\lambda\hat{k}\)
This matches (I) 3. The line is parallel to the z-axis through (0, 4, 0), so the foot from P is (0, 4, 12):
\(\sqrt{(3-0)^2+(4-4)^2+(12-12)^2} = 3\)