SATHEE: Parabola 1 Question 11

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

(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,

$x = t^{2} + t + 1$

$\Rightarrow x = {\frac{x - y}{2}}^{2} + \frac{x - y}{2} + 1$

$\Rightarrow 4 x = (x - y)^{2} + 2 x - 2 y + 4$

$\Rightarrow (x - y)^{2} = 2 (x + y - 2)$

$\Rightarrow x^{2} + y^{2} - 2 x y - 2 x - 2 y + 4 = 0$

Now, $Δ = 1 \cdot 1 \cdot 4 + 2 \cdot (- 1) (- 1) (- 1)$

$\begin{aligned} - 1 \times (- 1)^{2} - 1 \times (- 1)^{2} - 4 (- 1)^{2} \\ ∴ & Δ \end{aligned}$

$∴ Δ \neq 0$

Hence, it represents a equation of parabola.

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Parabola 1 Question 11

11. The curve described parametrically by x=t2+t+1,y=t2−t+1 represents

Assertion and Reason

Answer:

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11. The curve described parametrically by $x = t^{2} + t + 1, y = t^{2} - t + 1$ represents