Question

If the equation of the plane passing through the points $(4,5,1)$, $(3,9,4)$, $(-4,4,4)$ is $ax+by+cz=d$ then $a^4+b^3+c^2+d$ is equal to

Answer 1

Given points are $(4,5,1),(3,9,4),(-4,4,4)$
Now equation of plane passing through $(4,5,1)$ is given by,
$a(x-4)+b(y-5)+c(z-1)=0$
Also this plane passing through other two points.
$\Rightarrow -a+4b+3c=0$ and $-8a-b+3c=0$
Solving these, we get $a=5c/11,b=-7c/11$
Hence, equation of required plane is, $5(x-4)-7(y-5)+11(z-1)=0\Rightarrow 5x-7y+11z=-4$
Ergo $a^4+b^3+c^2+d=5^4+(-7{)}^3+{11}^2-4=499$