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MCQMediumJEE 2023Equation of Plane

JEE Mathematics 2023 Question with Solution

A plane PP contains the line of intersection of the plane r⃗.(i^+j^+k^)=6\vec{r}.(\hat{i}+\hat{j}+\hat{k}) = 6 and r⃗.(2i^+3j^+4k^)=−5\vec{r}.(2\hat{i}+3\hat{j}+4\hat{k}) = -5. If PP passes through the point (0,2,−2)(0, 2, -2), then the square of distance of the point (12,12,18)(12, 12, 18) from the plane PP is:

  • A

    620620

  • B

    12401240

  • C

    310310

  • D

    155155

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: A plane PP contains the line of intersection of the planes x+y+z−6=0x + y + z - 6 = 0 and 2x+3y+4z+5=02x + 3y + 4z + 5 = 0, and passes through (0,2,−2)(0, 2, -2).

Find: The square of the distance of (12,12,18)(12, 12, 18) from plane PP.

Any plane containing the line of intersection of the two given planes can be written as

(x+y+z−6)+λ(2x+3y+4z+5)=0(x + y + z - 6) + \lambda(2x + 3y + 4z + 5) = 0

Since it passes through (0,2,−2)(0, 2, -2), substitute this point:

(−6)+λ(6−8+5)=0(-6) + \lambda(6 - 8 + 5) = 0 −6+3λ=0-6 + 3\lambda = 0 λ=2\lambda = 2

Therefore, the equation of plane PP is

(x+y+z−6)+2(2x+3y+4z+5)=0(x + y + z - 6) + 2(2x + 3y + 4z + 5) = 0 5x+7y+9z+4=05x + 7y + 9z + 4 = 0

Now use the distance formula from the point (12,12,18)(12, 12, 18) to the plane:

d=∣5(12)+7(12)+9(18)+4∣52+72+92d = \frac{|5(12) + 7(12) + 9(18) + 4|}{\sqrt{5^2 + 7^2 + 9^2}} d=∣60+84+162+4∣25+49+81=310155d = \frac{|60 + 84 + 162 + 4|}{\sqrt{25 + 49 + 81}} = \frac{310}{\sqrt{155}}

Squaring,

d2=(310155)2=3102155=620d^2 = \left(\frac{310}{\sqrt{155}}\right)^2 = \frac{310^2}{155} = 620

Therefore, the square of the distance is 620620. The correct option is A.

Common mistakes

  • Using the normals of the given two planes directly as the normal of plane PP is incorrect. Plane PP belongs to the family of planes through their line of intersection, so write it as S1+λS2=0S_1 + \lambda S_2 = 0 and then use the given point to determine λ\lambda.

  • Substituting the point (0,2,−2)(0, 2, -2) incorrectly into 2x+3y+4z+52x + 3y + 4z + 5 is a common error. The correct value is 0+6−8+5=30 + 6 - 8 + 5 = 3, not any other number.

  • After finding the distance, forgetting that the question asks for the square of the distance leads to the wrong option. First compute dd, then evaluate d2d^2.

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