Question

The plane passing through the point $(-2,-2,-3)$ and containing the line joining the points $(1,1,1)$ and $(1,-1,2)$ makes an intercept $a,b,c$ along $x,y$ and $z$ axis respectively, then $\frac{1}{a}+\frac{1}{b}+\frac{a}{c}$ will be

• A. $3$
• B. $4$
• C. $6$
• D. $1$

Answer 1

Answer: (D)

$1$

The three points $(x_1,y_1,z_1)(x_2,y_2,z_2)$ and $(x_3,y_3,z_3)$ on plane are $(-2,-2,-3),(1,1,1)$ and $(1,-1,2)$.

$\therefore$ Equation of Plane $\Rightarrow$

$\left|\begin{array}{ccc}x-x_1& y-y_1& z-z_1\\ x_1-x_2& y_1-y_2& z_1-z_2\\ x_2-x_3& y_2-y_3& z_2-z_3\end{array}\right|=0\left|\begin{array}{ccc}x+2& y+2& z+3\\ 3& 3& 4\\ 3& 1& 5\end{array}\right|=0$

$\Rightarrow 11x-3y-6z=2$

$\therefore$ x-intercept $\Rightarrow \frac{2}{11}=a$

$\therefore$ y-intercept $\Rightarrow \frac{-2}{3}=b$

$\therefore$ z-intercept $\Rightarrow \frac{-1}{3}=c$

$\therefore \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$