Question

Consider the points $A(a,0,0),B(O,b,0),C(O,0,c)$, then which of the following is incorrect.

• A. Equation of plane through $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
• B. Area of the triangle $\Delta$ $ABC$ is $\frac{1}{2}\sqrt{{\left(ab\right)}^2+{\left(bc\right)}^2+{\left(ca\right)}^2}$
• C. Distance from $0(0,0,0)$ to the plane $ABC$ is $\frac{abc}{\sqrt{a^2{b}^2+b^2{c}^2+c^2{a}^2}}$
• D. None 'of these

Answer 1

The correct option is D None 'of these

(a) Equation of plane with intercepts a,b,c is given by,
$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
(b) area of triangle $ABC$ $=\sqrt{{\Delta }^{2}xy+{\Delta }^{2}yz+{\Delta }^{2}zx}=A$ (say)
where $\Delta xy=\frac{1}{2}\mid \left|\begin{array}{ccc}a& 0& 1\\ 0& b& 1\\ 0& 0& 1\end{array}\right|\mid =\frac{1}{2}ab$

$\Delta yz=\frac{1}{2}\mid \left|\begin{array}{ccc}0& 0& 1\\ b& 0& 1\\ 0& c& 1\end{array}\right|\mid =\frac{1}{2}bc$

and $\Delta yx=\frac{1}{2}\mid \left|\begin{array}{ccc}0& c& 1\\ a& 0& 1\\ 0& 0& 1\end{array}\right|\mid =\frac{1}{2}ac$
Hence, area of $\Delta ABC$ $=\frac{1}{2}\sqrt{a^2{b}^2+b^2{c}^2+c^2{a}^2}$
(c) distance from origin to the plane is,
$=\mid \frac{-1}{\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}}\mid$

$=\frac{abc}{\sqrt{a^2{b}^2+b^2{c}^2+c^2{a}^2}}$