Let the plane ax+by+cz=d pass through (2,3,-5) and be perpendicular to the planes 2x+y-5z=…

Question

Let the plane $ax+by+cz=d$ pass through $(2,3,-5)$ and be perpendicular to the planes $2x+y-5z=10$ and $3x+5y-7z=12$. If $a,b,c,d$ are integers, $d>0$, and $\gcd(|a|,|b|,|c|,d)=1$, then the value of $a+7b+c+20d$ is equal to:

Accepted Answer

Equation of pane through point (2, 3, -5) and perpendicular to planes 2x + y - 5z = 10 and 3x + 5y - 7z = 12 is

$$\left| {\begin{pmatrix} {x - 2} & {y - 3} & {z + 5}  \
   2 & 1 & { - 5}  \
   3 & 5 & { - 7}  \

\end{pmatrix} } \right| = 0$$

$$\therefore$$ Equation of plane is $(x - 2)( - 7 + 25) - (y - 3)$

$$( - 14 + 15) + (z + 5)\,.\,7 = 0$$

$$\therefore$$ $18x - y + 7z + 2 = 0$

$$ \Rightarrow 18x - y + 7z =  - 2$$

$$\therefore$$ $- 18x + y - 7z = 2$

On comparing with $ax + by + cz = d$ where d > 0 is a = - 18, b = 1, c = - 7, d = 2

$$\therefore$$ $a + 7b + c + 20d = 22$

Solutions · 1