If the plane r ˉ = ( λ + μ ) i ^ + ( 2 + μ ) j ^ + ( 3 λ + 2 μ ) k ^ , where λ and μ are parameters, intersects coordinate axes at points ( a , 0 , 0 ) , ( 0 , b , 0 ) and ( 0 , 0 , c ) then a + b +

Question

If the plane r ˉ = ( λ + μ ) i ^ + ( 2 + μ ) j ^ ​ + ( 3 λ + 2 μ ) k ^ , where λ and μ are parameters, intersects coordinate axes at points ( a , 0 , 0 ) , ( 0 , b , 0 ) and ( 0 , 0 , c ) then a + b + c = ...

TestHub · Accepted Answer

The parametric equations of the plane are given by:

$x = λ + μ$

$y = 2 + μ$

$z = 3 λ + 2 μ$

From the second equation, we have $μ = y - 2$ .

Substituting $μ$ into the first equation:

$x = λ + y - 2 \Rightarrow λ = x - y + 2$

Now, substituting both $λ$ and $μ$ into the third equation to eliminate the parameters:

$z = 3 (x - y + 2) + 2 (y - 2)$

$z = 3 x - 3 y + 6 + 2 y - 4$

$z = 3 x - y + 2$

Rearranging this, we get the Cartesian equation of the plane:

$3 x - y - z + 2 = 0$

To find the $x$ -intercept $a$ , put $y = 0$ and $z = 0$ :

$3 a - 0 - 0 + 2 = 0 \Rightarrow a = - \frac{2}{3}$

To find the $y$ -intercept $b$ , put $x = 0$ and $z = 0$ :

$0 - b - 0 + 2 = 0 \Rightarrow b = 2$

To find the $z$ -intercept $c$ , put $x = 0$ and $y = 0$ :

$0 - 0 - c + 2 = 0 \Rightarrow c = 2$

The sum of the intercepts is:

$a + b + c = - \frac{2}{3} + 2 + 2 = 4 - \frac{2}{3} = \frac{10}{3}$

Answer: $\frac{10}{3}$

Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

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