Question

Find the equation of the plane through the points $A(2,2-1),B(3,4,2)$ and $C(7,0,6)$

• A. $5x+2y-3z=17$
• B. $5x+2y+3z=17$
• C. $5x+y-3z=7$
• D. $5x+y+3z=7$

Answer 1

The correct option is A $5x+2y-3z=17$

The general equation of a plane through $(2,2-1)$ is

$a(x-2+b(y-2)+c(z+1)=0$ ...(1)

It will pass through $B(3,4,2)$ and $C(7,0,6)$ if

$a(3-2)+b(4-2)+c(2+1)=0\Rightarrow a+2b+3c=0$ ...(2)

and $a(7-2)+b(0-2)+c(6+1)=0\Rightarrow 5a-2b+7c=0$ ...(3)

Solving (2) and (3) by cross-multiplication, we have

$\frac{a}{14+6}=\frac{b}{15-7}=\frac{c}{-2-10}$

$\Rightarrow \frac{a}{5}=\frac{b}{2}=\frac{c}{-3}=\lambda$ (say)

$\Rightarrow a=5\lambda ,b=2\lambda$ and $c=-3\lambda$

Substituting the values of $a,b$ and $c$ in (1), we get

$5\lambda (x-2)+2\lambda (y-2)-3\lambda (z+1)=0\Rightarrow 5(x-2)+2(y-2)-3(z+1)=0\Rightarrow 5x+2y-3z=17$

which is the required equation of the plane.