Question

Find the intercepts of the plane \[5x - 3y + 6z = 60\] on the co-ordinate axes.
a) (10, 20, -10)
b) (10, -20, 12)
c) (12, -20, 10)
d) (12, 20, -10)

Answer 1

Hint:

Formula Used:Intercept form of a plane is given by \[\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1\].
 Where a, b and c are the x-intercept, y-intercept and z-intercept, respectively.


Complete step by step solution:Equation of the given plane is\[5x - 3y + 6z = 60\].
Let us rewrite the above equation in intercept form.
\[5x - 3y + 6z = 60\]
Divide by 60 on both sides.
             \[\dfrac{{5x - 3y + 6z}}{{60}} = \dfrac{{60}}{{60}}\]
             \[\dfrac{{5x}}{{60}} - \dfrac{{3y}}{{60}} + \dfrac{{6z}}{{60}} = 1\]
\[\dfrac{x}{{\left( {\dfrac{{60}}{5}} \right)}} + \dfrac{{3y}}{{\left( {\dfrac{{60}}{{ - 3}}} \right)}} + \dfrac{z}{{\left( {\dfrac{{60}}{6}} \right)}} = 1\]
             \[\dfrac{x}{{12}} + \dfrac{{3y}}{{ - 20}} + \dfrac{z}{{10}} = 1\]
This is of the form, \[\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1\].
x-intercept = a = 12
y-intercept = b = -20
z-intercept = c = 10
Hence the required answer is (12, -20. 10).



Option ‘C’ is correct



Note: This question can also be solved using the following method.
Equation of the plane is\[5x - 3y + 6z = 60\].
x-intercept:
In the x-axis, the value of y and z are zero.
In order to find x-intercept, substitute y = 0 and z = 0 in\[5x - 3y + 6z = 60\].
\[5x - 3\left( 0 \right) + 6\left( 0 \right) = 60\]
                       \[5x = 60\]
                        \[x = 12\]
So, the x-intercept is 12.
y-intercept:
In the y-axis, the value of x and z are zero.
In order to find y-intercept, substitute x = 0 and z = 0 in\[5x - 3y + 6z = 60\].
  \[5(0) - 3y + 6\left( 0 \right) = 60\]
                      \[ - 3y = 60\]
                          \[y = - 20\]
So, the y-intercept is -20.
z-intercept:
In the z-axis, the value of x and y are zero.
In order to find z-intercept, substitute x = 0 and y = 0 in\[5x - 3y + 6z = 60\].
  \[5(0) - 3\left( 0 \right) + 6z = 60\]
                       \[6z = 60\]
                         \[z = 10\]
So, the z-intercept is 10.
Hence the required answer is (12, -20. 10).
So, the correct choice is c.