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3D Geometry question

2020 · 7 Jan · Shift 2 · Q29

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3D Geometry
2020 · 7 Jan · Shift 2 · Q29

3D Geometry question

2020 · 7 Jan · Shift 2 · Q29

JEE MainMathematics3D GeometryNumerical+4 / −1

If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (

\alpha

, 7, 1) is

\left( {{5 \over 3},{7 \over 3},{{17} \over 3}} \right)

, then

\alpha

is equal to

\underline{\hspace{2cm}}

.

Numerical answer

View written solutionFree

Correct answer: 4

Given data

Point from which perpendicular is drawn: $P=(1,0,3)$
A point on the line: $A=(\alpha,7,1)$
Foot of perpendicular on the line: $F=\left(\frac53,\frac73,\frac{17}3\right)$

Since $F$ is the foot of the perpendicular from $P$ to the line, two facts are true:

$F$ lies on the line.
The direction vector of the line is perpendicular to $\overrightarrow{PF}$ .

Also, because both $A$ and $F$ lie on the line, the vector $\overrightarrow{AF}$ is a direction vector of the line.

Find $\overrightarrow{PF}$

$\overrightarrow{PF}=F-P=\left(\frac53-1,\frac73-0,\frac{17}3-3\right)$

$\overrightarrow{PF}=\left(\frac23,\frac73,\frac83\right)$

Find $\overrightarrow{AF}$

$\overrightarrow{AF}=F-A=\left(\frac53-\alpha,\frac73-7,\frac{17}3-1\right)$

$\overrightarrow{AF}=\left(\frac53-\alpha,-\frac{14}3,\frac{14}3\right)$

Use perpendicular condition

Since $\overrightarrow{PF}$ is perpendicular to the line, it is perpendicular to $\overrightarrow{AF}$ :

$\overrightarrow{PF}\cdot \overrightarrow{AF}=0$

So,

$\left(\frac23,\frac73,\frac83\right)\cdot \left(\frac53-\alpha,-\frac{14}3,\frac{14}3\right)=0$

Compute the dot product:

$\frac23\left(\frac53-\alpha\right)+\frac73\left(-\frac{14}3\right)+\frac83\left(\frac{14}3\right)=0$

$\frac{10}{9}-\frac{2\alpha}{3}-\frac{98}{9}+\frac{112}{9}=0$

$\frac{10-98+112}{9}-\frac{2\alpha}{3}=0$

$\frac{24}{9}-\frac{2\alpha}{3}=0$

$\frac{8}{3}-\frac{2\alpha}{3}=0$

Multiply by $3$ :

$8-2\alpha=0$

$2\alpha=8$

$\alpha=4$

Final answer

$\boxed{4}$

Comparison with stored correct answer

Stored correct answer = $4$

Our derived answer = $4$

Hence, the answer agrees with the stored correct answer.

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