Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

The coordinates of point P are $(a,b,c)$.

Since PA is perpendicular to the XY plane, the $z$-coordinate of A is $0$. Thus, the coordinates of A are $(a,b,0)$.

Since PB is perpendicular to the ZX plane, the $y$-coordinate of B is $0$. Thus, the coordinates of B are $(a,0,c)$.

The equation of the plane passing through the origin O $(0,0,0)$, A $(a,b,0)$, and B $(a,0,c)$ is given by:

$\left|\begin{array}{ccc}x& y& z\\ a& b& 0\\ a& 0& c\end{array}\right|=0$

Expanding the determinant along the first row, we get:

$x(bc-0)-y(ac-0)+z(0-ab)=0$

$bcx-acy-abz=0$

Dividing the entire equation by $abc$, we get:

$\frac{x}{a}-\frac{y}{b}-\frac{z}{c}=0$

Answer: $\frac{x}{a}-\frac{y}{b}-\frac{z}{c}=0$