Answer:
The required equations are
and
.
Step-by-step explanation:
The given parametric equation of the line,
, is
so, an arbitrary point on the line is ![R(x,y,z)=R(5+t, 6, -2-3t)](https://tex.z-dn.net/?f=R%28x%2Cy%2Cz%29%3DR%285%2Bt%2C%206%2C%20-2-3t%29)
The vector equation of the line passing through the points
and
is
![\vec P + \lambda \vec{(PR)}=0](https://tex.z-dn.net/?f=%5Cvec%20P%20%2B%20%5Clambda%20%5Cvec%7B%28PR%29%7D%3D0)
![\Rightarrow (-5 \hat i + 7 \hat j - 8 \hat k )+\lambda \left((5+t-(-5))\hat i + (6-7)\hat j +(-2-3t-8)\hat k\right)=0](https://tex.z-dn.net/?f=%5CRightarrow%20%28-5%20%5Chat%20i%20%2B%207%20%5Chat%20j%20-%208%20%5Chat%20k%20%29%2B%5Clambda%20%5Cleft%28%285%2Bt-%28-5%29%29%5Chat%20i%20%2B%20%286-7%29%5Chat%20j%20%2B%28-2-3t-8%29%5Chat%20k%5Cright%29%3D0)
![\Rightarrow (-5 \hat i + 7 \hat j - 8 \hat k )+\lambda \left((10+t)\hat i -\hat j +(6-3t)\hat k\right)=0\;\cdots (i)](https://tex.z-dn.net/?f=%5CRightarrow%20%28-5%20%5Chat%20i%20%2B%207%20%5Chat%20j%20-%208%20%5Chat%20k%20%29%2B%5Clambda%20%5Cleft%28%2810%2Bt%29%5Chat%20i%20-%5Chat%20j%20%2B%286-3t%29%5Chat%20k%5Cright%29%3D0%5C%3B%5Ccdots%20%28i%29)
The given equation can also be written as
![\frac {x-5}{1}=\frac {v-6}{0}=\frac{z+2}{-3}=t \; \cdots (ii)](https://tex.z-dn.net/?f=%5Cfrac%20%7Bx-5%7D%7B1%7D%3D%5Cfrac%20%7Bv-6%7D%7B0%7D%3D%5Cfrac%7Bz%2B2%7D%7B-3%7D%3Dt%20%5C%3B%20%5Ccdots%20%28ii%29)
The standard equation of the line passes through the point
and having direction
is
![\frac {x-x_0}{a_1}=\frac {y-y_0}{a_2}=\frac{z-z_0}{a_3}=t \;\cdots (iii)](https://tex.z-dn.net/?f=%5Cfrac%20%7Bx-x_0%7D%7Ba_1%7D%3D%5Cfrac%20%7By-y_0%7D%7Ba_2%7D%3D%5Cfrac%7Bz-z_0%7D%7Ba_3%7D%3Dt%20%5C%3B%5Ccdots%20%28iii%29)
Here, The value of the parameter,
, of any point
at a distance
from the point,
, can be determined by
![|t \vec v|=d\;\cdots (iv)](https://tex.z-dn.net/?f=%7Ct%20%5Cvec%20v%7C%3Dd%5C%3B%5Ccdots%20%28iv%29)
Comparing equations
and ![(iii)](https://tex.z-dn.net/?f=%28iii%29)
The line is passing through the point
having direction
.
Note that the point
is the same as
obtained above.
Now, the value of the parameter,
, for point
at distance
from the point
can be determined by equation
, we have
![|t(\hat i -3 \hat k)|=3](https://tex.z-dn.net/?f=%7Ct%28%5Chat%20i%20-3%20%5Chat%20k%29%7C%3D3)
![\Rightarrow t^2|(\hat i -3 \hat k)|^2=9](https://tex.z-dn.net/?f=%5CRightarrow%20t%5E2%7C%28%5Chat%20i%20-3%20%5Chat%20k%29%7C%5E2%3D9)
![\Rightarrow 10t^2=9](https://tex.z-dn.net/?f=%5CRightarrow%2010t%5E2%3D9)
![\Rightarrow t^2=\frac {9}{10}](https://tex.z-dn.net/?f=%5CRightarrow%20t%5E2%3D%5Cfrac%20%7B9%7D%7B10%7D)
![\Rightarrow t=\pm \frac {3}{\sqrt {10}}](https://tex.z-dn.net/?f=%5CRightarrow%20t%3D%5Cpm%20%5Cfrac%20%7B3%7D%7B%5Csqrt%20%7B10%7D%7D)
Put the value of
in equation
, the required equations are as follows:
For ![t= \frac {3}{\sqrt {10}}](https://tex.z-dn.net/?f=t%3D%20%5Cfrac%20%7B3%7D%7B%5Csqrt%20%7B10%7D%7D)
![(-5 \hat i + 7 \hat j - 8 \hat k )+\lambda \left((10+\frac {3}{\sqrt {10}})\hat i -\hat j +\left(6-3\times \frac {3}{\sqrt {10}})\hat k\right)=0](https://tex.z-dn.net/?f=%28-5%20%5Chat%20i%20%2B%207%20%5Chat%20j%20-%208%20%5Chat%20k%20%29%2B%5Clambda%20%5Cleft%28%2810%2B%5Cfrac%20%7B3%7D%7B%5Csqrt%20%7B10%7D%7D%29%5Chat%20i%20-%5Chat%20j%20%2B%5Cleft%286-3%5Ctimes%20%5Cfrac%20%7B3%7D%7B%5Csqrt%20%7B10%7D%7D%29%5Chat%20k%5Cright%29%3D0)
![\Rightarrow (-5 \hat i + 7 \hat j - 8 \hat k )+\lambda \left((10+\frac {3}{\sqrt {10}})\hat i -\hat j +(6- \frac {9}{\sqrt {10}})\hat k\right)=0](https://tex.z-dn.net/?f=%5CRightarrow%20%28-5%20%5Chat%20i%20%2B%207%20%5Chat%20j%20-%208%20%5Chat%20k%20%29%2B%5Clambda%20%5Cleft%28%2810%2B%5Cfrac%20%7B3%7D%7B%5Csqrt%20%7B10%7D%7D%29%5Chat%20i%20-%5Chat%20j%20%2B%286-%20%5Cfrac%20%7B9%7D%7B%5Csqrt%20%7B10%7D%7D%29%5Chat%20k%5Cright%29%3D0)
and for
,
![(-5 \hat i + 7 \hat j - 8 \hat k )+\lambda \left((10+\left (-\frac {3}{\sqrt {10}}\right))\hat i -\hat j +(6-3\times \left(-\frac {3}{\sqrt {10}}\right)\hat k\right)=0](https://tex.z-dn.net/?f=%28-5%20%5Chat%20i%20%2B%207%20%5Chat%20j%20-%208%20%5Chat%20k%20%29%2B%5Clambda%20%5Cleft%28%2810%2B%5Cleft%20%28-%5Cfrac%20%7B3%7D%7B%5Csqrt%20%7B10%7D%7D%5Cright%29%29%5Chat%20i%20-%5Chat%20j%20%2B%286-3%5Ctimes%20%5Cleft%28-%5Cfrac%20%7B3%7D%7B%5Csqrt%20%7B10%7D%7D%5Cright%29%5Chat%20k%5Cright%29%3D0)
![\Rightarrow (-5 \hat i + 7 \hat j - 8 \hat k )+\lambda \left((10-\frac {3}{\sqrt {10}})\hat i -\hat j +(6+ \frac {9}{\sqrt {10}})\hat k\right)=0](https://tex.z-dn.net/?f=%5CRightarrow%20%20%28-5%20%5Chat%20i%20%2B%207%20%5Chat%20j%20-%208%20%5Chat%20k%20%29%2B%5Clambda%20%5Cleft%28%2810-%5Cfrac%20%7B3%7D%7B%5Csqrt%20%7B10%7D%7D%29%5Chat%20i%20-%5Chat%20j%20%2B%286%2B%20%5Cfrac%20%7B9%7D%7B%5Csqrt%20%7B10%7D%7D%29%5Chat%20k%5Cright%29%3D0)