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3D Geometry question

2019 · 8 Apr · Shift 1 · Q35
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3D Geometry question

2019 · 8 Apr · Shift 1 · Q35

JEE MainMathematics3D GeometryMCQ+4 / −1
The length of the perpendicular from the point (2, –1, 4) on the straight line, x+310{{x + 3} \over {10}}10x+3​= y−2−7{{y - 2} \over {-7}}−7y−2​=z1{{z} \over {1}}1z​ is :
  1. A
    less than 2
  2. B
    greater than 4
  3. C
    greater than 2 but less than 3
  4. D
    greater than 3 but less than 4
View written solutionFree

Correct answer: D

  1. Write the line in parametric form

The line is x+310=y−2−7=z1=t\frac{x+3}{10}=\frac{y-2}{-7}=\frac{z}{1}=t10x+3​=−7y−2​=1z​=t

So, x=−3+10t,y=2−7t,z=tx=-3+10t,\qquad y=2-7t,\qquad z=tx=−3+10t,y=2−7t,z=t

A point on the line is A(−3,2,0)A(-3,2,0)A(−3,2,0)

and its direction vector is d⃗=(10,−7,1)\vec d=(10,-7,1)d=(10,−7,1)

The given point is P(2,−1,4)P(2,-1,4)P(2,−1,4)

  1. Use distance from a point to a line in 3D

Distance from point PPP to the line through AAA with direction vector d⃗\vec dd is Distance=∣AP⃗×d⃗∣∣d⃗∣\text{Distance}=\frac{|\vec{AP}\times \vec d|}{|\vec d|}Distance=∣d∣∣AP×d∣​

First, AP⃗=P−A=(2−(−3),−1−2,4−0)=(5,−3,4)\vec{AP}=P-A=(2-(-3),-1-2,4-0)=(5,-3,4)AP=P−A=(2−(−3),−1−2,4−0)=(5,−3,4)

  1. Compute the cross product
\begin{vmatrix} \hat i & \hat j & \hat k \\ 5 & -3 & 4 \\ 10 & -7 & 1 \end{vmatrix}$$ $$=\hat i\big((-3)(1)-4(-7)\big)-\hat j\big(5(1)-4(10)\big)+\hat k\big(5(-7)-(-3)(10)\big)$$ $$=\hat i(-3+28)-\hat j(5-40)+\hat k(-35+30)$$ $$=25\hat i+35\hat j-5\hat k$$ So, $$\vec{AP}\times \vec d=(25,35,-5)$$ Its magnitude is $$|\vec{AP}\times \vec d|=\sqrt{25^2+35^2+(-5)^2} =\sqrt{625+1225+25}= \sqrt{1875}=25\sqrt{3}$$ 4. **Compute the magnitude of direction vector** $$|\vec d|=\sqrt{10^2+(-7)^2+1^2}= \sqrt{100+49+1}= \sqrt{150}=5\sqrt{6}$$ 5. **Find the perpendicular distance** $$\text{Distance}=\frac{25\sqrt{3}}{5\sqrt{6}}=5\cdot\frac{\sqrt{3}}{\sqrt{6}}=5\sqrt{\frac{1}{2}}=\frac{5}{\sqrt2}$$ $$\text{Distance}=\frac{5\sqrt2}{2}\approx 3.54$$ 6. **Match with the options** Since $$3<3.54<4$$ the correct option is: **D: greater than 3 but less than 4**
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