The distance of the point

Question:

The distance of the point $(1,-2,3)$ from the plane $x-y+z=5$ measured parallel to a line, whose direction ratios are $2,3,-6$ is :

  1. 3

  2. 5

  3. 2

  4. 1


Correct Option: , 4

Solution:

$(1+2 \lambda)+2-3 \lambda+3-6 \lambda=5$

$\Rightarrow 6-7 \lambda=5 \Rightarrow \lambda=\frac{1}{7}$

so, $P=\left(\frac{9}{7},-\frac{11}{7}, \frac{15}{7}\right)$

$\mathrm{AP}=\sqrt{\left(1-\frac{9}{7}\right)^{2}+\left(-2+\frac{11}{7}\right)^{2}+\left(3-\frac{15}{7}\right)^{2}}$

$\mathrm{AP}=\sqrt{\left(\frac{4}{49}\right)+\frac{9}{49}+\frac{36}{49}}=1$

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