Question

The curve x = t² + t + 1,y = t² – t + 1 represents

Options

• a parabola

• a hyperbola

• an ellipse

• a rectangular hyperbola

Answer 1

an ellipse

Explanation:

Given `x = t^2 + t + 1`  .....(i)

`y = t^2 - t + 1`  .....(ii)

∴ `x + y = 2(1 + t^2)`   .....(iii)

`x - y = 2t`  .....(iv)

Now, from equations (iii) and (iv), we get

`x + y = 2[1 + ((x - y)/2)^2]`

⇒ `x^2 + y^2 - 2xy - 2x - 2y + 4` = 0  ......(v)

On comparing with `ax^2 + 2hxy + by2 + 2gx + 2fy + c` = 0

We get, `a = 1, b = 1, c = 4, h = -1, g = - 1, f = -1`

Δ = `abc + 2fgh = af^2 - bg^2 - ch^2`

= `1.1.4 + 2(-1)(-1)(-1) - 1(-1)^2 - 1(-1)^2 - 4(-1)^2`

= `4 - 2 - 1 - 1 - 4 = - 4 = 0`

and `ab - h^2 = 1.1 - (-1)^2 = 1 - 1` = 0

So, it is equation of a parabola.