Question
The curve described parametrically by $$x = {t^2} + t + 1,\,\,y = {t^2} - t + 1$$ represents-
A.
a pair of straight lines
B.
an ellipse
C.
a parabola
D.
a hyperbola
Answer :
a parabola
Solution :
KEY CONCEPT :
The equation $$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$$
represents a parabola if $$\Delta \ne 0$$ and $${h^2} = ab$$
where $$\Delta = abc + 2fgh - a{f^2} - b{g^2} - c{h^2}$$
Now we have $$x = {t^2} + t + 1$$ and $$y = {t^2} - t + 1$$
$$\frac{{x + y}}{2} = {t^2} + 1,\,\frac{{x - y}}{2} = t$$ ( Adding and subtracting values of $$x$$ and $$y$$ )
Eliminating $$t,\,\,\,2\left( {x + y} \right) = {\left( {x - y} \right)^2} + 4.....(1)$$
$$ \Rightarrow {x^2} - 2xy + {y^2} - 2x - 2y + 4 = 0.....(2)$$
Here, $$a = 1,\,h = - 1,\,b = 1,\,g = - 1,\,f = - 1,\,c = 4$$
$$\therefore \Delta \ne 0.\,\,{\text{and }}{h^2} = ab$$
Hence the given curve represents a parabola.