"If for some $\alpha$ and $\beta$ in R, the intersection of the following three places x + 4y – 2z = 1 x + 7y – 5z = b x + 5y + $\alpha$z = 5 is a line in R3, then $\alpha$ + $\beta$ is equal to :"

Question

Accepted Answer

"For planes to intersect on a line there should be infinite solution of the given system of equations.

For infinite solutions

$\Delta$ = $$\left| {\matrix{ 1 & 4 & { - 2} \cr 1 & 7 & { - 5} \cr 1 & 5 & \alpha \cr } } \right|$$ = 0

$\Rightarrow$ 1(7$\alpha$ + 25) – 4($\alpha$ + 5) – 2(5 – 7) = 0

$\Rightarrow$ 7$\alpha$ + 25 – 4$\alpha$ – 20 + 4 = 0

$\Rightarrow$ 3$\alpha$ + 9 = 0

$\Rightarrow$ $\alpha$ = -3

Also $\Delta$_z = 0

$\Rightarrow$ $$\left| {\matrix{ 1 & 4 & 1 \cr 1 & 7 & \beta \cr 1 & 5 & 5 \cr } } \right|$$ = 0

$\Rightarrow$ 1(35 – 5$\beta$) – 4(5 – $\beta$) + 1(5 – 7) = 0

$\Rightarrow$ 35 - 5$\beta$ - 20 + 4$\beta$ - 2 = 0

$\Rightarrow$ $\beta$ = 13

$\therefore$ $\alpha$ + $\beta$ = -3 + 13 = 10"

If for some $\alpha$ and $\beta$ in R, the intersection of the following three places
x + 4y – 2z = 1
x + 7y – 5z = b
x + 5y + $\alpha$z = 5
is a line in R³, then $\alpha$ + $\beta$ is equal to :

Solution

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If for some $\alpha$ and $\beta$ in R, the intersection of the following three places x + 4y – 2z = 1 x + 7y – 5z = b x + 5y + $\alpha$z = 5 is a line in R3, then $\alpha$ + $\beta$ is equal to :

Solution

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More Matrices and Determinants questions

Drill 25 more like these. Every day. Free.

If for some $\alpha$ and $\beta$ in R, the intersection of the following three places
x + 4y – 2z = 1
x + 7y – 5z = b
x + 5y + $\alpha$z = 5
is a line in R³, then $\alpha$ + $\beta$ is equal to :