JEE 2023 Mathematics PYQ — Let the plane pass through the intersection of the planes and and… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
Let the plane P pass through the intersection of the planes 2x+3y−z=2 and x+2y+3z=6 and be perpendicular to the plane 2x+y−z=0. If d is the distance of P from the point (−7,1,1), then d2 is equal to:
Choose the correct answer:
A.
83250
(Correct Answer)
B.
82250
C.
5315
D.
8325
Correct Answer:
83250
Explanation
Solution
Step 1: Write the equation of the family of planes
The equation of a plane passing through the intersection of two planes P1=0 and P2=0 is given by P1+λP2=0.
(2x+3y−z−2)+λ(x+2y+3z−6)=0
Rearranging the terms:
(2+λ)x+(3+2λ)y+(−1+3λ)z−(2+6λ)=0…(Eq. 1)
Step 2: Use the perpendicularity condition
The plane P is perpendicular to the plane 2x+y−z=0. The dot product of their normal vectors must be zero:
(2+λ)(2)+(3+2λ)(1)+(−1+3λ)(−1)=0
4+2λ+3+2λ+1−3λ=0
λ+8=0⟹λ=−8
Step 3: Substitute λ back into Eq. 1
(2−8)x+(3−16)y+(−1−24)z−(2−48)=0
−6x−13y−25z+46=0
Multiplying by −1:
6x+13y+25z−46=0
Step 4: Calculate the distance d from point (−7,1,1)
The formula for the distance of a point (x1,y1,z1) from a plane ax+by+cz+d=0 is:
d=a2+b2+c2∣ax1+by1+cz1+d∣
d=62+132+252∣6(−7)+13(1)+25(1)−46∣
d=36+169+625∣−42+13+25−46∣
d=830∣−50∣=83050
Step 5: Find d2
d2=(83050)2=8302500=83250
Correct Option: (1)
Explanation
Solution
Step 1: Write the equation of the family of planes
The equation of a plane passing through the intersection of two planes P1=0 and P2=0 is given by P1+λP2=0.
(2x+3y−z−2)+λ(x+2y+3z−6)=0
Rearranging the terms:
(2+λ)x+(3+2λ)y+(−1+3λ)z−(2+6λ)=0…(Eq. 1)
Step 2: Use the perpendicularity condition
The plane P is perpendicular to the plane 2x+y−z=0. The dot product of their normal vectors must be zero:
(2+λ)(2)+(3+2λ)(1)+(−1+3λ)(−1)=0
4+2λ+3+2λ+1−3λ=0
λ+8=0⟹λ=−8
Step 3: Substitute λ back into Eq. 1
(2−8)x+(3−16)y+(−1−24)z−(2−48)=0
−6x−13y−25z+46=0
Multiplying by −1:
6x+13y+25z−46=0
Step 4: Calculate the distance d from point (−7,1,1)
The formula for the distance of a point (x1,y1,z1) from a plane ax+by+cz+d=0 is:
