Parabola 1 Question 11
11. 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
Assertion and Reason
Show Answer
Answer:
Correct Answer: 11. (c)
Solution:
- Given curves are $x=t^{2}+t+1$
and
$$ y=t^{2}-t+1 $$
On subtracting Eq. (ii) from Eq. (i),
Thus,
$$ x=t^{2}+t+1 $$
$\Rightarrow \quad x=\frac{x-y}{2}^{2}+\frac{x-y}{2}+1$
$$ \Rightarrow \quad 4 x=(x-y)^{2}+2 x-2 y+4 $$
$$ \Rightarrow \quad(x-y)^{2}=2(x+y-2) $$
$\Rightarrow \quad x^{2}+y^{2}-2 x y-2 x-2 y+4=0$
Now, $\Delta=1 \cdot 1 \cdot 4+2 \cdot(-1)(-1)(-1)$
$$ \begin{aligned} & \quad-1 \times(-1)^{2}-1 \times(-1)^{2}-4(-1)^{2} \\ \therefore & \quad \Delta \end{aligned} $$
$$ \therefore \quad \Delta \neq 0 $$
Hence, it represents a equation of parabola.
