Question

A variable plane is at a constant distance \(p\) from the origin \(\mathrm{O}\) and meets the set of rectangular axes \(\mathrm{OX}_{i}\) \((i=1,2,3)\) at points \(\mathrm{A}_{i}\) \((i=1,2,3)\), respectively. If planes are drawn through \(\mathrm{A}_{1}\), \(\mathrm{A}_{2}\), \(\mathrm{A}_{3}\), which are parallel to the coordinate planes, then the locus of their point of intersection is :

• A. \(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}=\frac{1}{p}\)
• B. \(\frac{1}{x_{1}^{2}}+\frac{1}{x_{2}^{2}}+\frac{1}{x_{3}^{2}}=\frac{1}{p^{2}}\)
• C. \(\frac{1}{x_{1}^{3}}+\frac{1}{x_{2}^{3}}+\frac{1}{x_{3}^{3}}=\frac{1}{p^{3}}\)
• D. \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=p^{2}\)

Hint:

Write the plane in intercept form \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\); its distance from the origin gives a relation among \(a,b,c\).

Answer 1

The Correct Option is B

Let the plane be \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\), meeting the axes at \(A_{1}(a,0,0)\), \(A_{2}(0,b,0)\), \(A_{3}(0,0,c)\).
Distance from the origin is \(p\): \(\frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}}}=p\Rightarrow \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{p^{2}}\).
The planes through \(A_{1},A_{2},A_{3}\) parallel to the coordinate planes are \(x=a\), \(y=b\), \(z=c\); they meet at \((x_{1},x_{2},x_{3})=(a,b,c)\).
Substituting \(a=x_{1},b=x_{2},c=x_{3}\): \(\frac{1}{x_{1}^{2}}+\frac{1}{x_{2}^{2}}+\frac{1}{x_{3}^{2}}=\frac{1}{p^{2}}\).
Answer: (2) \(\frac{1}{x_{1}^{2}}+\frac{1}{x_{2}^{2}}+\frac{1}{x_{3}^{2}}=\frac{1}{p^{2}}\)