JEE Main 2020 · 9 January · morning · Single correct · +4 / −1
If for some α and β in R, the intersection of the following three places
x + 4y – 2z = 1
x + 7y – 5z = b
x + 5y + αz = 5
is a line in R³, then α + β is equal to :
- (a) -10
- (b) 0
- (c) 10
- (d) 2
Answer and solution
Answer (c) 10
Solution
Maths sir · 4 steps
Three planes share a whole line only when the third plane is built from the first two. Find that mix.
- Line condition. Three planes meet in a line, and not a single point, only when one equation is a combination of the other two. So write the third plane as λ times the first plus μ times the second. The b in the second equation is β.
x + 5y + α z - 5 = λ(x + 4y - 2z - 1) + μ(x + 7y - 5z - β)
- Match x and y. From x: λ + μ = 1. From y: 4λ + 7μ = 5, which becomes 4 + 3μ = 5. So μ = 1/3 and λ = 2/3.
- Match z and constant. From z: α = -2λ - 5μ = -4/3 - 5/3 = -3. From the constants: 5 = λ + μβ = 2/3 + β/3, so β = 13.
- So. α + β = -3 + 13 = 10, which is option (c).
The idea to keep. Planes meeting in a line means the system has infinitely many solutions. One plane must be λ P₁ + μ P₂. Match the coefficients first, then match the constant terms.
Answer from the NTA final answer key. Solution by PrepWiser.