21.  Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{p^{2}}.

Answers (1)

The equation of plane having a, b and c intercepts with x, y  and z-axis respectively is given by
\frac{x}{a}+\frac{y}{b}+\frac{z}{c}= 1
The distance p of the plane from the origin is given by 
\\p = \left | \frac{\frac{0}{a}+\frac{0}{b}+\frac{0}{c}-1}{\sqrt{(\frac{1}{a})^2+(\frac{1}{b})^2(\frac{1}{c})^2}} \right |\\ \\ p = \left | \frac{-1}{\sqrt{(\frac{1}{a})^2+(\frac{1}{b})^2(\frac{1}{c})^2}} \right |\\ \\ \frac{1}{p^2}= \frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}
Hence proved 

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