Question

If a line with direction ratio 2 : 2 : 1 intersects the line \[\dfrac{{{\text{x - 7}}}}{3}{\text{ = }}\dfrac{{{\text{y - 5}}}}{2}{\text{ = }}\dfrac{{{\text{z - 3}}}}{1}\] and \[\dfrac{{{\text{x - 1}}}}{2}{\text{ = }}\dfrac{{{\text{y + 1}}}}{4}{\text{ = }}\dfrac{{{\text{z + 1}}}}{3}\] at A and B then AB =
A. $\sqrt 2 $ units
B. 2 units
C. $\sqrt 3 $ units
D. 3 units

Answer 1

Hint: To solve this question, we will find the value of point A and B by letting \[\dfrac{{{\text{x - 7}}}}{3}{\text{ = }}\dfrac{{{\text{y - 5}}}}{2}{\text{ = }}\dfrac{{{\text{z - 3}}}}{1}\] = $\lambda $ and \[\dfrac{{{\text{x - 1}}}}{2}{\text{ = }}\dfrac{{{\text{y + 1}}}}{4}{\text{ = }}\dfrac{{{\text{z + 1}}}}{3}\] = $\mu $. Then we will find the direction ratios of AB and compare it to the given direction ratios. We will use the distance formula to find AB.

Complete step-by-step answer:

Now, the line having direction ratio 2: 2: 1 intersect the line \[\dfrac{{{\text{x - 7}}}}{3}{\text{ = }}\dfrac{{{\text{y - 5}}}}{2}{\text{ = }}\dfrac{{{\text{z - 3}}}}{1}\] at point A. So, we will find the coordinates of point A and as both lines intersect, so A should satisfy both the lines.
Let \[\dfrac{{{\text{x - 7}}}}{3}{\text{ = }}\dfrac{{{\text{y - 5}}}}{2}{\text{ = }}\dfrac{{{\text{z - 3}}}}{1}\] = $\lambda $. So, we get
${\text{x - 7 = 3}}\lambda $. So, ${\text{x = 3}}\lambda {\text{ + }}{\text{ 7}}$. Similarly, we get \[{\text{y = 2}}\lambda {\text{ + 5}}\] and \[{\text{z = }}\lambda {\text{ + 3}}\]. So, coordinates of point A are ($3\lambda {\text{ + 7}}$, $2\lambda {\text{ + 5}}$, $\lambda {\text{ + 3}}$). Also, the line intersects \[\dfrac{{{\text{x - 1}}}}{2}{\text{ = }}\dfrac{{{\text{y + 1}}}}{4}{\text{ = }}\dfrac{{{\text{z + 1}}}}{3}\] at point B.
So, let \[\dfrac{{{\text{x - 1}}}}{2}{\text{ = }}\dfrac{{{\text{y + 1}}}}{4}{\text{ = }}\dfrac{{{\text{z + 1}}}}{3}\] = $\mu $. Proceeding exactly the same in the above case, we get the coordinates of point B. So, coordinates of point B are ($2\mu {\text{ + 1}}$, $4\mu {\text{ - 1}}$, $3\mu {\text{ - 1}}$).
Now, direction ratios of line AB = ($2\mu {\text{ + 1}}$ - ($3\lambda {\text{ + 7}}$), $4\mu {\text{ - 1}}$ - ($2\lambda {\text{ + 5}}$), $3\mu {\text{ - 1}}$ - ($\lambda {\text{ + 3}}$)). So, direction ratios of AB are ($2\mu {\text{ - 3}}\lambda {\text{ - 6}}$, $4\mu {\text{ - 2}}\lambda {\text{ - 6}}$, $3\mu {\text{ - }}\lambda {\text{ - 4}}$). Now, as the points A and B are lying on the line having direction ratio 2: 2: 1. So, the direction ratios of AB will equal to direction ratios of line.
So, we get \[\dfrac{{2\mu {\text{ - 3}}\lambda {\text{ - 6}}}}{2}{\text{ = }}\dfrac{{4\mu {\text{ - 2}}\lambda {\text{ - 6}}}}{2}{\text{ = }}\dfrac{{3\mu {\text{ - }}\lambda {\text{ - 4}}}}{1}\]. So, we get
$2\mu {\text{ - 3}}\lambda {\text{ - 6 = 4}}\mu {\text{ - 2}}\lambda {\text{ - 6}}$
$ \Rightarrow $ $\lambda {\text{ = - 2}}\mu $ … (1)
Also, $2\mu {\text{ - 3}}\lambda {\text{ - 6 = 6}}\mu {\text{ - 2}}\lambda {\text{ - 8}}$
$ \Rightarrow $ $4\mu {\text{ + }}\lambda {\text{ - 2 = 0}}$
Putting $\lambda {\text{ = - 2}}\mu $ from equation (1), we get
$4\mu {\text{ - 2}}\mu {\text{ - 2 = 0}}$ $ \Rightarrow $ $\mu {\text{ = 1}}$
so, from equation (1), we get $\lambda {\text{ = - 2}}$. Putting these values to find the value of A and B, we get
A = (1, 1, 1) and B = (3, 3, 2)
Now, we will use the distance formula to find AB. Distance formula is given by
D = $\sqrt {{{\left( {{{\text{x}}_2}{\text{ - }}{{\text{x}}_1}} \right)}^2}{\text{ + (}}{{\text{y}}_2}{\text{ - }}{{\text{y}}_1}{)^2}{\text{ + (}}{{\text{z}}_2}{\text{ - }}{{\text{z}}_1}{)^2}} $, where D is the distance between two given points.
So, applying the values in the above formula, we get
AB = $\sqrt {{{\left( {{\text{3 - 1}}} \right)}^2}{\text{ + (3 - 1}}{)^2}{\text{ + (2 - 1}}{)^2}} $ = $\sqrt {{2^2}{\text{ + }}{{\text{2}}^2}{\text{ + }}{{\text{1}}^2}} $
So, AB = $\sqrt 9 $ = 3units
So, option (D) is correct.
Note: When we come up with such types of questions, we have to find the value of intersecting points from the given equations of lines. Then we find the direction ratios of the points found and compare it to the given direction ratios. After it we will find the value of the distance asked by using the distance formula.