ParabolaHard
Question
The curve described parametrically by x = t2 + t +1, y = t2 - t +1 represents
Options
A.a pair of straght lines
B.an ellipse
C.a parabola
D.a hyperbola
Solution
Given curves are x = t2 + t + 1 ......(i)
and y = t2 - t + 1 ......(ii)
On subtracting Eq. (ii) from Eq. (ii),
x - y = 2t
Thus, x = t2 + t + 1
⇒
⇒ 4x = (x - y)2 + 2x - 2y + 4
⇒ (x - y)2 = 2(x + y - 2)
⇒ x2 + y2 - 2xy - 2x - 2y + 4 = 0
Now, ᐃ = 1.1.4 + 2.(-1)(-1)(-1)
-1 ×(-1)2 -1 ×(-1)2 - 4(-1)2
= 4 - 2 -1-1- 4 = -4
∴ ᐃ ≠ 0
and ab - h2 = 1.1 - (-1)2 = 1 - 1 = 0
Hence, it represents a rquation of parabola.
and y = t2 - t + 1 ......(ii)
On subtracting Eq. (ii) from Eq. (ii),
x - y = 2t
Thus, x = t2 + t + 1
⇒
⇒ 4x = (x - y)2 + 2x - 2y + 4
⇒ (x - y)2 = 2(x + y - 2)
⇒ x2 + y2 - 2xy - 2x - 2y + 4 = 0
Now, ᐃ = 1.1.4 + 2.(-1)(-1)(-1)
-1 ×(-1)2 -1 ×(-1)2 - 4(-1)2
= 4 - 2 -1-1- 4 = -4
∴ ᐃ ≠ 0
and ab - h2 = 1.1 - (-1)2 = 1 - 1 = 0
Hence, it represents a rquation of parabola.
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