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

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Answer:

Correct Answer: 11. (c)

Solution:

  1. 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.



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