The values of t such that the origin lies inside or on the circle is [- 0. 786, + 0.786]
For what values of t does the vertex of the parabola lie inside or on the circle
The standard equation of the circle has the following form:
(t - h)² + (y - k)² = r² (1)
Where:
- (h, k) - Coordinates of the center.
- r - Radius of the circle.
In accordance with the statement, the following condition has to be met in (1):
(h, k) = (t₁, t₁²)
(t, y) = (t₂, t₂²)
r = 1
There are two cases:
- (h, k) = (0, 0)
- (t, y) = (0, 0)
Case 1
t₂² + t₂⁴ = 1
t₂⁴ + t₂² - 1 = 0
(t₂ - 0.786) · (t₂ + 0.786) · (t₂ - i 1.272) · (t₂ + i 1.272) = 0
t₂ = ± 0.786
Case 2
t₁² + t₁⁴ = 1
t₁⁴ + t₁² - 1 = 0
(t₁ - 0.786) · (t₁ + 0.786) · (t₁ - i 1.272) · (t₁ + i 1.272) = 0
t₁ = ± 0.786
Then, the values of t such that the origin lies inside or on the circle is [- 0. 786, + 0.786]
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