Logo
Jeeify
MCQMediumJEE 2023Equation of Plane

JEE Mathematics 2023 Question with Solution

Let the plane P:4x−y+z=10P : 4x - y + z = 10 be rotated by an angle π2\frac{\pi}{2} about its line of intersection with the plane x+y−z=4x + y - z = 4. If α\alpha is the distance of the point (2,3,−4)(2, 3, -4) from the new position of the plane PP, then 35α35\alpha is:

  • A

    9090

  • B

    8585

  • C

    105105

  • D

    126126

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: The original plane is 4x−y+z−10=04x - y + z - 10 = 0 and the axis of rotation is its line of intersection with the plane x+y−z−4=0x + y - z - 4 = 0.

Find: The value of 35α35\alpha, where α\alpha is the distance of the point (2,3,−4)(2,3,-4) from the new plane.

Any plane passing through the intersection of the two planes can be written as

(4x−y+z−10)+λ(x+y−z−4)=0(4x - y + z - 10) + \lambda(x + y - z - 4) = 0

So the new plane is taken as

(4x−y+z−10)+λ(x+y−z−4)=0(4x - y + z - 10) + \lambda(x + y - z - 4) = 0

Using the condition shown in the solution, we solve for λ\lambda:

4(4+λ)−1(−1+λ)+1(1−λ)=04(4 + \lambda) - 1(-1 + \lambda) + 1(1 - \lambda) = 0

Expanding,

16+4λ+1−λ+1−λ=016 + 4\lambda + 1 - \lambda + 1 - \lambda = 0

Therefore,

18+2λ=018 + 2\lambda = 0

So,

λ=−9\lambda = -9

Substituting back,

(4x−y+z−10)−9(x+y−z−4)=0(4x - y + z - 10) - 9(x + y - z - 4) = 0

This simplifies to

−5x−10y+10z+26=0-5x - 10y + 10z + 26 = 0

Now the distance of the point (2,3,−4)(2,3,-4) from this plane is

α=5415\alpha = \frac{54}{15}

Hence,

35α=5415×35=12635\alpha = \frac{54}{15} \times 35 = 126

Therefore, the correct option is D.

Common mistakes

  • Taking the family of planes incorrectly. The new plane must pass through the line of intersection of the two given planes, so its equation should be of the form P1+λP2=0P_1 + \lambda P_2 = 0. Using any unrelated plane equation loses the rotation axis.

  • Using the point-to-plane distance formula with the wrong denominator. For a plane ax+by+cz+d=0ax+by+cz+d=0, the denominator is a2+b2+c2\sqrt{a^2+b^2+c^2}, not a+b+ca+b+c. Always use the magnitude of the normal vector.

  • Substituting λ=−9\lambda = -9 incorrectly while simplifying the plane equation. Sign errors in expanding −9(x+y−z−4)-9(x+y-z-4) can change the final plane and produce a wrong distance. Expand each term carefully.

Practice more Equation of Plane questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trialMore on Equation of Plane

Related questions