Two system of rectangular axes have the same origin. If a plane cuts them at distances, $a$ , $b$ , $c$ and $a_{1}$ , $b_{1}$ , $c_{1}$ from the origin, then

\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} = \frac{1}{a_{1}^{2}} + \frac{1}{b_{1}^{2}} + \frac{1}{c_{1}^{2}}

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\frac{1}{a^{2}} - \frac{1}{b^{2}} + \frac{1}{c^{2}} = \frac{1}{a_{1}^{2}} - \frac{1}{b_{1}^{2}} + \frac{1}{c_{1}^{2}}

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a^{2} + b^{2} + c^{2} = a_{1}^{2} + b_{1}^{2} + c_{1}^{2}

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a^{2} - b^{2} + c^{2} = a_{1}^{2} - b_{1}^{2} + c_{1}^{2}

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Solution

The correct option is B $\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} = \frac{1}{a_{1}^{2}} + \frac{1}{b_{1}^{2}} + \frac{1}{c_{1}^{2}}$
Let the equation of the plane be
$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ and $\frac{x}{a_{1}} + \frac{y}{b_{1}} + \frac{z}{c_{1}} = 1$
$a x + b y + c z + d = 0 p e r p e n d i c u l a r d i s t a n c e f r o m o r i g i n i s \frac{| d |}{\sqrt{a^{2} + b^{2} + c^{2}}}$
as they have the same origin their perpendicular distance is constant.
$\frac{1}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}} = \frac{1}{\sqrt{\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}}}$
$\frac{1}{{a_{1}}^{2}} + \frac{1}{{b_{1}}^{2}} + \frac{1}{{c_{1}}^{2}} = \frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}$

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Similar questions

(a_{1}, b_{1}) a n d (a_{2}, b_{2}) i s (a_{1} - a_{2}) x + (b_{1} - b_{2}) y + c = 0

{[{(a_{1} {+ a}_{2} + . . . . a_{n})}^{2} {(b_{1} + b_{2} + . . . . + b_{n})}^{2}]}^{1 / 2} < \sqrt{{a_{1}}^{2} + {b_{1}}^{2} +} \sqrt{{a_{2}}^{2} + {b_{2}}^{2}} + . . . . + \sqrt{{a_{n}}^{2} + {b_{n}}^{2}}

a_{r}, b_{r} (r = 1, 2, ., n)

a = a_{1} i + a_{2} j + a_{3} k, b = b_{1} i + b_{2} j + b_{3} k

c = c_{1} i + c_{2} j + c_{3} k

\frac{π}{6}

{∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣}^{2}

The condition for $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ to be consistent and have a unique solution is

a_{1} x^{2} + b_{1} x + c_{1} = 0, a_{2} x^{2} + b_{2} x + c_{2} = 0

a_{3} x^{2} + b_{3} x + c_{3} = 0

\frac{c_{1} a_{2} + c_{2} a_{1}}{c_{1} a_{2} - c_{2} a_{1}} + \frac{a_{1} a_{2} b_{3}}{a_{3} (a_{1} b_{2} - a_{2} b_{1})} = 0

{(\frac{a_{1} b_{2} - a_{2} b_{1}}{a_{1} c_{2} - a_{2} c_{1}})}^{2} = \frac{a_{1} a_{2} c_{3}}{c_{1} c_{2} a_{3}}

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