JEE 2023 Mathematics PYQ — Let the equation of the plane containing the line be and the dist… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
Let the equation of the plane P containing the line x+10=28−y=z be ax+by+3z=2(a+b) and the distance of the plane P from the point (1,27,7) be c. Then a2+b2+c2 is equal to
Choose the correct answer:
A.
355
(Correct Answer)
B.
354
C.
353
D.
352
Correct Answer:
355
Explanation
1. Simplify the Line Equation
The given line is:
x+10=−2y−8=z
From this, we identify:
A point on the line (A):(−10,8,0)
Direction vector of the line (v):(1,−2,1)
2. Determine a and b
Since the plane ax+by+3z=2(a+b) contains the line:
Step A: Substitute point (−10,8,0) into the plane equation:
−10a+8b+3(0)=2a+2b
−12a+6b=0
b=2a
Step B: Use the direction vector (1,−2,1):
The normal vector to the plane is n=(a,b,3). Since the line lies in the plane, n⋅v=0:
a(1)+b(−2)+3(1)=0
a−2b+3=0
Substitute b=2a into the equation:
a−2(2a)+3=0
a−4a+3=0⟹−3a=−3⟹a=1
Since b=2a, then b=2.
The Plane Equation:
1x+2y+3z=2(1+2)⟹x+2y+3z=6
x+2y+3z−6=0
3. Calculate the Distance c
The distance c from the point (1,27,7) to the plane x+2y+3z−6=0 is:
c=12+22+32∣(1)+2(27)+3(7)−6∣
c=14∣1+54+21−6∣
c=1470
c2=144900=350
4. Final Calculation
We have a=1, b=2, and c2=350.
a2+b2+c2=12+22+350
a2+b2+c2=1+4+350=355
Final Answer:
a2+b2+c2=355
Explanation
1. Simplify the Line Equation
The given line is:
x+10=−2y−8=z
From this, we identify:
A point on the line (A):(−10,8,0)
Direction vector of the line (v):(1,−2,1)
2. Determine a and b
Since the plane ax+by+3z=2(a+b) contains the line:
Step A: Substitute point (−10,8,0) into the plane equation:
−10a+8b+3(0)=2a+2b
−12a+6b=0
b=2a
Step B: Use the direction vector (1,−2,1):
The normal vector to the plane is n=(a,b,3). Since the line lies in the plane, n⋅v=0:
a(1)+b(−2)+3(1)=0
a−2b+3=0
Substitute b=2a into the equation:
a−2(2a)+3=0
a−4a+3=0⟹−3a=−3⟹a=1
Since b=2a, then b=2.
The Plane Equation:
1x+2y+3z=2(1+2)⟹x+2y+3z=6
x+2y+3z−6=0
3. Calculate the Distance c
The distance c from the point (1,27,7) to the plane x+2y+3z−6=0 is:
