If $\displaystyle p_1$ and $\displaystyle p_2$ are the lengths of the perpendiculars from the points $\displaystyle \left ( 2,3,4 \right )$ and $\displaystyle \left ( 1,1,4 \right )$ respectively from the plane $\displaystyle 3x-6y+2z+11=0$, then $\displaystyle p_1$, $\displaystyle p_2$ are the roots of the equation

Using the perpendicular distance formula from a point to a plane, we calculate p1 from (2,3,4) as |3(2) - 6(3) + 2(4) + 11| / sqrt(3^2 + (-6)^2 + 2^2) = 1. We then calculate p2 from (1,1,4) as |3(1) - 6(1) + 2(4) + 11| / 7 = 16/7. Since the required roots are p1 = 1 and p2 = 16/7, the quadratic equation is (p - 1)(p - 16/7) = 0. Multiplying out the terms gives p^2 - 23/7 p + 16/7 = 0, and multiplying by 7 yields 7p^2 - 23p + 16 = 0.

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