Let the line L pass through the point (3,5,2) and make equal angles with the positive coordinate axes. If the distance of L from the point (2,r,1) is , then the sum of all possible values o

Question

Let the line L pass through the point (−3,5,2) and make equal angles with the positive coordinate axes. If the distance of L from the point (−2,r,1) is $\sqrt{\frac{14}{3}}$ , then the sum of all possible values of r is : [2026]

Smart Achievers · Accepted Answer

(4)

$Equation of line is : \frac{x + 3}{1} = \frac{y - 5}{1} = \frac{z - 2}{1} = λ$

$∴$ $General point R on line is R (λ - 3, λ + 5, λ + 2)$

$\vec{P R} \equiv (λ - 1, λ + 5 - r, λ + 1)$

Now $\vec{P R} \cdot \vec{d} = 0$

$\Rightarrow (λ - 1) 1 + (λ + 5 - r) 1 + (λ + 1) 1 = 0$

$\Rightarrow 3 λ - r + 5 = 0$

$\Rightarrow λ = \frac{r - 5}{3}$

$∴$ $R \equiv (\frac{r - 5}{3} - 3, \frac{r - 5}{3} + 5, \frac{r - 5}{3} + 2)$

$R \equiv (\frac{r - 14}{3}, \frac{r + 10}{3}, \frac{r + 1}{3})$

Now

$P R = \sqrt{\frac{14}{3}} \Rightarrow {(P R)}^{2} = \frac{14}{3}$

$\Rightarrow {(\frac{r - 14}{3} + 2)}^{2} + {(\frac{r + 10}{3} - r)}^{2} + {(\frac{r + 1}{3} - 1)}^{2} = \frac{14}{3}$

$\Rightarrow \frac{{(r - 8)}^{2}}{9} + \frac{{(10 - 2 r)}^{2}}{9} + \frac{{(r - 2)}^{2}}{9} = \frac{14}{3}$

$\Rightarrow (r^{2} - 16 r + 64) + (100 + 4 r^{2} - 40 r) + (r^{2} - 4 r + 4) = 42$

$\Rightarrow 6 r^{2} - 60 r + 126 = 0$

$\Rightarrow r^{2} - 10 r + 21 = 0$

$\Rightarrow r = 3, 7$

$Sum of possible values of r is 10$

Solution

Q.
Let the line L pass through the point (−3,5,2) and make equal angles with the positive coordinate axes. If the distance of L from the point (−2,r,1) is $\sqrt{\frac{14}{3}}$ , then the sum of all possible values of r is : [2026]

1 6

2 16

3 12

4 10

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