If the points (1,1,) and (-3,0,1) are equidistant from the plane 3x+4y-12z+13=0, then sati…

Question

If the points $(1,1,\lambda)$ and $(-3,0,1)$ are equidistant from the plane $3x+4y-12z+13=0$, then $\lambda$ satisfies the equation:

Accepted Answer

$$|3+4-12\lambda+13|=|-9+0-12+13|\implies|20-12\lambda|=8\implies \lambda=1\text{ or }\tfrac73.$$
Quadratic with these roots: $3(\lambda-1)\left(\lambda-\tfrac73\right)=3\lambda^2-10\lambda+7=0$.

Solutions · 1