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Q.

The line $L : \frac{x - 2}{1} = \frac{y + 2}{- 3} = \frac{z - 5}{2}$ intersect the plane $2 x - 3 y + 4 z = 163$ at point P and intersect the yz plane at point Q. If the length of PQ is $a \sqrt{b}$ where $a, b \in N$ and $a > 3,$ then value of (b – a) is equal to _____

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answer is 5.

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Detailed Solution

$\begin{array}{l} \frac{x - 2}{1} = \frac{y + 2}{- 3} = \frac{z - 5}{2} = λ \cdot \\ Let P (λ + 2, - 3 λ - 2, 2 λ + 5) l i e s i n, 2 x - 3 y + 4 z = 163 \\ \Rightarrow 2 (λ + 2) - 3 (- 3 λ - 2) + 4 (2 λ + 5) = 163 \\ \Rightarrow λ = 7 \Rightarrow P (9, - 23, 19) \end{array}$

since line intersect YZ plane i.e x = 0

$\begin{array}{l} \Rightarrow λ + 2 = 0 \Rightarrow λ = - 2 \Rightarrow Q (0, 4, 1) \\ \Rightarrow PQ = \sqrt{9^{2} + 27^{2} + 18^{2}} = 9 \sqrt{14} \\ b - a = 5 \end{array}$

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The line L:x−21=y+2−3=z−52 intersect the plane 2x−3y+4z=163 at point P and intersect the yz plane at point Q. If the length of PQ is ab where a,b∈N and a>3, then value of (b – a) is equal to _____