Question

In a three- dimensional xyz space, the equation \[{x^2} - 5x + 6 = 0\] represents:
A. Points
B. Planes
C. curves
D. Pair of straight lines

Answer 1

Hint: We use factorization method to solve for the values of ‘x’ from the quadratic equation. Since all other coordinates except ‘x’ are zero we check the equations formed from the obtained value are the equations of plan, curve or straight lines.
* General equation of a plane is \[ax + by + cz - d = 0\]
* General equation of a curve depends on the types of curves, parabola, hyperbola etc.
* General equation of straight line is \[ax + by = c\]

Complete step by step solution:
We are given a three-dimensional XYZ space.
This means there are three axes i.e. X-axis, Y-axis and Z-axis.
We are given the equation \[{x^2} - 5x + 6 = 0\]
Now we factorize the middle part of the equation to find the possible roots.
We can write \[ - 5x = - 3x - 2x\]
\[ \Rightarrow {x^2} - 3x - 2x + 6 = 0\]
Take ‘x’ common from first two terms and ‘-2’ common from last two terms
\[ \Rightarrow x(x - 3) - 2(x - 3) = 0\]
Take common the factor that is repeated and group the other remaining factor
\[ \Rightarrow (x - 3)(x - 2) = 0\]
Now we know the equation is equal to zero if either one of the values in the product is zero or both are zero.
We equate each factor to zero
\[ \Rightarrow x - 3 = 0\] and \[x - 2 = 0\]
Now we compare these two equations with general equations of plane, curve and straight lines.
Since we can write \[x - 3 = 0\] and \[x - 2 = 0\] as \[x + 0y + 0z - 3 = 0\] and \[x + 0y + 0z - 2 = 0\]
These become equations of planes.
\[\therefore \]The equation represents planes.

\[\therefore \]Option B is correct.

Note: Students are likely to make the mistake of choosing the “pair of straight lines” as the correct option. Keep in mind we cannot choose straight lines as an option because here we are looking at a three dimensional space and we have straight lines in a two dimensional space. So avoid writing that \[x = 2,x = 3\] are a pair of straight lines.