find the image of the point ( 1,6,3) inthe line x/1 = y-1/2 = z-2/3 Also write the equation - Maths - Three Dimensional Geometry

Question

find the image of the point ( 1,6,3) inthe line x/1 = y-1/2 =
z-2/3 Also write the equation of the line joining the given point
and its image and find the length of the segment joining the given
point and its image

Gursheen kaur. · Accepted Answer

Let P (1, 6, 3) be the point whose image is to be determined.

Draw perpendicular from P to the given line which cuts the line L, which is its foot of perpendicular.

Then, image of P will be in The downward direction and at the same distance from L as P.

Let P' (x₁, y₁, z₁) be the image of point P. Then we will find the values of x₁, y₁, z₁.

Then,

Given line is

$\frac{x - 0}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} = λ \Rightarrow x = λ, y = 2 λ + 1, z = 3 λ + 2$

i.e., the co-ordinate of any general point which lies on the given line is (λ, 2λ + 1, 3λ + 2).

∴ Co-ordinate of L are (λ, 2λ + 1, 3λ + 2).

Now,

Direction ratio's of PL will be

λ – 1, 2λ + 1 – 6, 3λ + 2 – 3 = λ – 1, 2λ – 5, 3λ – 1

Also,

Direction ratio's of given line are 1, 2, 3.

But, PL is perpendicular to the given line.

∴ 1 (λ – 1) + 2 (2λ – 5) + 3 (3λ – 1) = 0

⇒ λ – 1 + 4λ – 10 + 9λ – 3 = 0

⇒ 14λ – 14 = 0

$\Rightarrow λ = 1$

∴ Co-ordinate of L are (1, 3, 5)

Now,

L is the mid point of PP',

By mid point formula

$\frac{1 + x_{1}}{2} = 1, \frac{6 + y_{1}}{2} = 3, \frac{3 + z_{1}}{2} = 5 \Rightarrow x_{1} = 2 - 1, y_{1} = 6 - 6, z_{1} = 10 - 3 \Rightarrow x_{1} = 1, y_{1} = 0, z_{1} = 7$

Hence, the image of (1, 6, 3) is (1, 0, 7).

150

find the image of the point ( 1,6,3) inthe line x/1 = y-1/2 = z-2/3. Also write the equation of the line joining the given point and its image and find the length of the segment joining the given point and its image.