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((14 / 3)), then the sum of all possible values of r is :

Equation line is : (x+3 / 1)=(y-5 / 1)=(z-2 / 1)=lambda ∴ General point R on line is R(lambda-3, lambda+5, lambda+2) \ -> PR equiv(lambda-1, lambda+5- r, lambda+1) Now -> PR * -> d=0 => (lambda-1) 1+(lambda+5- r) 1+(lambda+1) 1=0 => 3 lambda- r+5=0 => lambda=frac r-53 Therefore R equiv(frac r-53-3, frac r-53+5, frac r-53+2) R equiv(frac r-143, frac r+103, frac r+13) Now \ PR=sqrt((14 / 3)) => ( PR)^2=(14 / 3) => (frac r-143+2)^2+(frac r+103- r)^2+(frac r+13-1)^2=(14 / 3) => frac( r-8)^29+frac(10-2 r)^29+frac( r-2)^29=(14 / 3) => (r^2-16 r+64)+(100+4 r^2-40 r)+(r^2-4 r+4)=42 => 6 r^2-60 r+126=0 => r^2-10 r+21=0 => r=3,7 sum of possible value of r is =10

Let the line L pass through the point 352 and make equa... | null (Mathematics)

Equation line is :

\frac{x+3}{1}=\frac{y-5}{1}=\frac{z-2}{1}=\lambda\\

∴ General point R on line is

\mathrm{R}(\lambda-3, \lambda+5, \lambda+2)\\

\\ \overrightarrow{\mathrm{PR}} \equiv(\lambda-1, \lambda+5-\mathrm{r}, \lambda+1)\\

Now

\overrightarrow{\mathrm{PR}} \cdot \overrightarrow{\mathrm{d}}=0\\

\Rightarrow(\lambda-1) 1+(\lambda+5-\mathrm{r}) 1+(\lambda+1) 1=0\\

\Rightarrow 3 \lambda-\mathrm{r}+5=0\\

\Rightarrow \lambda=\frac{\mathrm{r}-5}{3}\\

\therefore \mathrm{R} \equiv\left(\frac{\mathrm{r}-5}{3}-3, \frac{\mathrm{r}-5}{3}+5, \frac{\mathrm{r}-5}{3}+2\right)\\

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

Now

\\ \mathrm{PR}=\sqrt{\frac{14}{3}} \Rightarrow(\mathrm{PR})^{2}=\frac{14}{3}\\

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

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

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

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

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

\Rightarrow \mathrm{r}=3,7\\

sum of possible value of r is =10

Explanation

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