Question

What are represented by the equation \[xy + {a^2} = a(x + y)\]

Answer 1

Hint: in this question, we have to find what is the geometrical nature or geometrical shape of given equation. In order to find that we just rearrange the given equation and compare it with various standard equations like straight line, parabola and ellipse etc.

Formula Used:\[a(c + d) = ac + ad\]
Where a, b, c, d are any number or variable.



Complete step by step solution:Given: \[xy + {a^2} = a(x + y)\]
Now we have to rearrange the given equation in order to get standard equation.
\[xy + {a^2} = ax + ay\]
\[xy + {a^2} - ax - ay = 0\]
\[xy - ax - ay + {a^2} = 0\]
\[x(y - a) - a(y - a) = 0\]
On simplification we get
\[(x - a)(y - a) = 0\]
The above equation represents the equation of pair of straight line.
Hence given equation is an equation of straight line.
hence given equation is an equation of pair of straight line. :< /b>



Note: In this type of questions always rearrange the given equation and try to arrange it in any standard form. After that, always compare the rearranged equation with all standard equations.
Don’t try to apply any formula related to any geometrical shape in this type of questions otherwise it becomes very complicated to get the solution.
Some standard equations are given as
Equation of straight lines
\[y = mx + c\]
Equation of parabola
\[{y^2} = 4ax\]
Equation of circle
\[{x^2} + {y^2} + 2gx + 2fy + c = 0\]
Equation of ellipse
\[\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1\]