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Definition of Vector

Let the plane...

Question

Let the plane $a x + b y + c z + d = 0$ bisect the line joining the points $(4, - 3, 1)$ and $(2, 3, - 5)$ at the right angles. If $a, b, c, d$ are integers, then the minimum value of $(a^{2} + b^{2} + c^{2} + d^{2})$ is

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Solution

Normal of plane $= - - \to P Q = - 2^i + 6^j - 6^k$
$a = - 2, b = 6, c = - 6$
Equation of plane is
$- 2 x + 6 y - 6 z + d = 0$

Mid point of $P (4, - 3, 1)$ and $Q (2, 3, - 5)$ is $M (3, 0, - 2) .$
Since, plane passes through $M,$
$∴ d = - 6$
So, equation of plane is
$- 2 x + 6 y - 6 z - 6 = 0$
$\Rightarrow x - 3 y + 3 z + 3 = 0$
$(a^{2} + b^{2} + c^{2} + d^{2})_{min} = 1^{2} + 9 + 9 + 9 = 28$

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Q. If the intercepts made on the axes by the plane which bisects the line joining the points

(1, 2, 3)

(- 3, 4, 5)

at right angles are

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