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Plane containing two given lines, then a distance from a point

This type of question, asked 3 times: 2021 Paper 19, 26 Jun 2022, 26 Jul 2022. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 19

    85
    Let QQ be the foot of the perpendicular from the point P(7,−2,13)P(7, - 2,13) on the plane containing the lines x+16=y−17=z−38\frac{x+ 1}{6}=\frac{y- 1}{7}=\frac{z- 3}{8} and x−13=y−25=z−37\frac{x- 1}{3}=\frac{y- 2}{5}=\frac{z- 3}{7} . Then (PQ)2(PQ)^{2}, is equal to

  2. Question 2

    26 Jun 2022

    165
    If the lines r→=(i^−j^+k^)+λ(3j^−k^)\overrightarrow{r}= (\widehat{i}-\widehat{j}+\widehat{k}) + \lambda(3\widehat{j}-\widehat{k}) and r→=(αi^−j^)+μ(2i^−3k^)\overrightarrow{r}= (\alpha\widehat{i}-\widehat{j}) + \mu(2\widehat{i}- 3\widehat{k}) are co-planar, then distance of the plane containing these two lines from the point (α,0,0)(\alpha,0,0) is :

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  3. Question 3

    26 Jul 2022

    172
    The largest value of aa, for which the perpendicular distance of the plane containing the lines r→=(i^+j^)+λ(i^+aj^−k^)\overrightarrow{r}= (\widehat{i}+\widehat{j}) + \lambda(\widehat{i}+ a\widehat{j}-\widehat{k}) and r→=(i^+j^)+μ(−i^+j^−ak^)\overrightarrow{r}= (\widehat{i}+\widehat{j}) + \mu( -\widehat{i}+\widehat{j}- a\widehat{k}) from the point (2,1,4)(2,1,4) is 3\sqrt{3}, is