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Question:
In $R^3$, consider the planes $P_1 : y = 0$ and $P_2 : x + z = 1$. Let $P_3$ be a plane, different from $P_1$ and $P_2$, which passes through the intersection of $P_1$ and $P_2$. If the distance of the point $(0, 1, 0)$ from $P_3$ is $1$ and the distance of a point $(\alpha, \beta, \gamma)$ from $P_3$ is $2$, then which of the following relations is (are) true ?
In $R^3$, consider the planes $P_1 : y = 0$ and $P_2 : x + z = 1$. Let $P_3$ be a plane, different from $P_1$ and $P_2$, which passes through the intersection of $P_1$ and $P_2$. If the distance of the point $(0, 1, 0)$ from $P_3$ is $1$ and the distance of a point $(\alpha, \beta, \gamma)$ from $P_3$ is $2$, then which of the following relations is (are) true ?
Updated On: Jun 14, 2022
- $2a + ? + 2? + 2 = 0$
- $2a - ? + 2? + 4 = 0$
- $2a + ? - 2? - 10 = 0$
- $2a - ? + 2? - 8 = 0$
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The Correct Option is D
Solution and Explanation
Equation of $P_3$ will be
$\left(x + z - 1\right) + ?y = 0$
$x + ?y + z - 1 = 0$
Its disantace from $\left(0, 1, 0\right)$ will be
$\left|\frac{0+\lambda+0-1}{\sqrt{1+\lambda^{2}+1}}\right| = 1$
$? \left(? - 1\right)^{2} = 1 + ?^{2} + 1$
$? ?^{2} - 2? + 1 = ?^{2} + 2$
$? ? = -1/2$
$?$ Equation of $P_{3}$ is $2x - y + 2z - 2 = 0$
Its distance from $\left(a, ?, ?\right)$ is
$\frac{\left|2a - ? + 2? - 2\right|}{3} = 2$
$2a - ? + 2? - 2 = \pm\, 6$
$? 2a - ? + 2? - 8 = 0$
and $2a - ? + 2? + 4 = 0$
$\left(x + z - 1\right) + ?y = 0$
$x + ?y + z - 1 = 0$
Its disantace from $\left(0, 1, 0\right)$ will be
$\left|\frac{0+\lambda+0-1}{\sqrt{1+\lambda^{2}+1}}\right| = 1$
$? \left(? - 1\right)^{2} = 1 + ?^{2} + 1$
$? ?^{2} - 2? + 1 = ?^{2} + 2$
$? ? = -1/2$
$?$ Equation of $P_{3}$ is $2x - y + 2z - 2 = 0$
Its distance from $\left(a, ?, ?\right)$ is
$\frac{\left|2a - ? + 2? - 2\right|}{3} = 2$
$2a - ? + 2? - 2 = \pm\, 6$
$? 2a - ? + 2? - 8 = 0$
and $2a - ? + 2? + 4 = 0$
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