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Question: What are represented by the equation \({{x}^{3}}-{{y}^{3}}=\left( y-a \right)\left( {{x}^{2}}-{{y}^{...

What are represented by the equation x3−y3=(y−a)(x2−y2){{x}^{3}}-{{y}^{3}}=\left( y-a \right)\left( {{x}^{2}}-{{y}^{2}} \right).

Explanation

Solution

Hint: Substitute the basic equations in place of (x3−y3)\left( {{x}^{3}}-{{y}^{3}} \right)and(x2−y2)\left( {{x}^{2}}-{{y}^{2}} \right).
Simplify them and equate them to zero. You will get the expression for a straight line and parabola.

Complete step-by-step answer:
We are given the equation
x3−y3=(y−a)(x2−y2)……(1).{{x}^{3}}-{{y}^{3}}=\left( y-a \right)\left( {{x}^{2}}-{{y}^{2}} \right)\ldots \ldots (1).
We know the basic expressions
x3−y3=(x−y)(x2+y2+xy) x2−y2=(x−y)(x+y) \begin{aligned} & {{x}^{3}}-{{y}^{3}}=\left( x-y \right)\left( {{x}^{2}}+{{y}^{2}}+xy \right) \\\ & {{x}^{2}}-{{y}^{2}}=\left( x-y \right)\left( x+y \right) \\\ \end{aligned}
Substitute these in equation (1).
(x−y)(x2+y2+xy)=(y−a)(x−y)(x+y)\left( x-y \right)\left( {{x}^{2}}+{{y}^{2}}+xy \right)=\left( y-a \right)\left( x-y \right)\left( x+y \right)
From this, x−y=0x-y=0

& \Rightarrow {{x}^{2}}+{{y}^{2}}+xy=\left( y-a \right)\left( x-y \right)\ldots \ldots (2). \\\ \end{aligned}$$ Here we have expanded the value of $( y-a) (x-y)$. $\Rightarrow\left( y-a \right)\left( x-y \right)=xy-{{y}^{2}}-ax+ay \\\ $ Thus substitute the value of $$\left( y-a \right)\left( x+y \right)$$ in equation (2). $$\Rightarrow {{x}^{2}}+{{y}^{2}}+xy=xy+{{y}^{2}}-ax-ay$$ Now cancel $$xy$$ and $${{y}^{2}}$$from LHS and RHS of the above expression and rearrange the equations. $$\begin{aligned} & {{x}^{2}}=-ay-ax \\\ & \Rightarrow {{x}^{2}}+ax+ay=0 \\\ \end{aligned}$$ Thus we obtained 2 curves, $$x-y=0$$ and $${{x}^{2}}+ax+ay=0$$ $$x-y=0$$, represents a straight line. $${{x}^{2}}+ax+ay=0$$, represents a parabola. Thus, the given equation represents a straight line and parabola. Note: You might cancel out $$\left( x-y \right)$$ in the initial stages of the problem. Remember that $$x-y=0$$represents the equation of a straight line. Remember the basic formulas that we have used here. Without the basic equations you can't simplify the given expression. The equation $${{x}^{2}}+ax+ay=0$$ is said to be a quadratic equation as the degree of equation is 2. When we draw a graph for this equation, we get a parabola. The graph of a quadratic equation in two variables (y = ax2 \+ bx + c ) is called a parabola and the equation $$x-y=0$$ is said to be a linear equation in one variable as the degree of equation is 2. When we draw a graph for this equation, we get a straight line.