Question

If a variable plane passes through a fixed point (1, - 2, 3) and meets the coordinate axes at points A, B, C, then the point of intersection of the planes through A, B, C parallel to the coordinate planes lies on

Answer 1

Let the equation of variable plane be $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
As this plane passes through (1,-2,3) , hence this point should satisfy the plane .
So $\frac{1}{a}+\frac{-2}{b}+\frac{3}{c}=1$ (1)

As the equation is in intercept form , so its intercept on axes are A ( a ,0,0 ) B(0,b,0 ) and C(0 ,0 ,c )
As the plane passes through A ,B and C , parallel to coordinate planes .
So the planes are x = a , y = b and z = c

so replace a,b,c by x , y and z in (1) we get the required plane
So the equation is $\frac{1}{x}+\frac{-2}{y}+\frac{3}{c}=1$