Name: Equation of a Line in 3D (Symmetric Form)
Uploaded: 2019-07-19
Duration: 11 min 8 s
Description: We have seen the equation of a line in 2D. Here, in this video, we will see the equation of a line in 3D and also see why vectors play an important role in 3...

Question

Find the equation of the sphere which has centre at the origin and touches the line

2 (x + 1) = 2 - y = z + 3 .

Accepted Answer

Consider a sphere having centre at origin $O$ and touches the line $2 (x + 1) = 2 - y = z + 3$ at point $N$ .

Let the equation of sphere has centre at origin $O (0, 0, 0)$ is ${(x - 0)}^{2} + {(y - 0)}^{2} + {(z - 0)}^{2} = r^{2}$

Now sphere touches the line $2 (x + 1) = 2 - y = z + 3$

$\Rightarrow \frac{x + 1}{\frac{1}{2}} = \frac{y - 2}{- 1} = \frac{z + 3}{1} = λ$ (say)

Now coordinates of any point say point of contact $N = (\frac{λ}{2} - 1, - λ + 2, λ - 3)$

Now direction ratios of line are $(\frac{1}{2}, - 1, 1)$

Also, direction ratios of line $O N = (\frac{λ}{2} - 1, - λ + 2, λ - 3)$

Since line $O N$ is perpendicular to given line.

$\Rightarrow (\frac{λ}{2} - 1) (\frac{1}{2}) - 1 (- λ + 2) + 1 (λ - 3) = 0$

$\Rightarrow \frac{λ}{4} - \frac{1}{2} + λ - 2 + λ - 3 = 0$

$\Rightarrow λ = \frac{22}{9}$

Therefore, the coordinates of $N = (\frac{2}{9}, - \frac{4}{9}, - \frac{5}{9})$

Now length of $O N$ = radius of sphere = $\sqrt{{(\frac{2}{9} - 0)}^{2} + {(- \frac{4}{9} - 0)}^{2} + {(- \frac{5}{9} - 0)}^{2}} = \sqrt{\frac{4}{81} + \frac{16}{81} + \frac{25}{81}}$

Therefore, equation of sphere becomes

${(x - 0)}^{2} + {(y - 0)}^{2} + {(z - 0)}^{2} = \frac{4}{81} + \frac{16}{81} + \frac{25}{81}$

$\Rightarrow x^{2} + y^{2} + z^{2} = \frac{5}{9}$

$\Rightarrow 9 (x^{2} + y^{2} + z^{2}) = 5$

$\Rightarrow 9 (x^{2} + y^{2} + z^{2}) = 5$

Find the equation of the sphere which has centre at the origin and touches the line $2 (x + 1) = 2 - y = z + 3 .$

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

Find the equation of the sphere which has centre at the origin and touches the line 2x+1=2-y=z+3 .

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

Find the equation of the sphere which has centre at the origin and touches the line $2 (x + 1) = 2 - y = z + 3 .$