Question

A plane passes through the point $(3,5,7)$. If the direction ratios of its normal is equal to the intercept made by plane $x+3y+2z=9$ with the coordinate axes, then the equation of the plane is ?

Answer 1

The plane passes through $(3,5,7)$

equation of plane $\Rightarrow ax+by+cz=d$.

$(a,b,c)$ are direction returns of its normal

$3a+5b+7c=d$

Intercepts made by plane $x+3y+2=a$ with coordinates area.

$(a,0,0)a+0+0=9$

$\Rightarrow a=9$

$(0,b,0)0+3\left(b\right)+2\left(0\right)=9$

$\Rightarrow b=3$

$(0,0,c)0+3\left(b\right)+2\left(0\right)=9$

$\Rightarrow c=9/2$

$\therefore (a,b,c)\to (9,3,9/2)\to$ dr's of the normal to

using $\left(1\right)$

$3\left(9\right)+5\left(3\right)+7\left(9\right)=d$

$27+15+\frac{63}{2}=d$

$\frac{147}{2}=d$

$\therefore$ equation of the plane $\Rightarrow ax+by+cz=d$

$\Rightarrow 9x+3y+\frac{9}{2}z=\frac{147}{2}$

$\Rightarrow 18x+6y+9z=147$

equation of required plane $\Rightarrow 6x+2y+37=49$