The point `(t^(2)+2t+5,2t^(2)+t-2)` lies on the line `x+y=2` for

Options

A. All real values of t
B. Some real values of t
C. `t=(-3+-sqrt(3))/6`
D. None of these

Answer

To determine the values of \( t \) for which the point \( (t^2 + 2t + 5, 2t^2 + t - 2) \) lies on the line \( x + y = 2 \), we follow these steps:

### Step 1: Substitute the coordinates into the line equation
We know that the point lies on the line if it satisfies the equation \( x + y = 2 \). Here, we have:
- \( x = t^2 + 2t + 5 \)
- \( y = 2t^2 + t - 2 \)

Substituting these into the equation gives us:
\[ (t^2 + 2t + 5) + (2t^2 + t - 2) = 2 \]

### Step 2: Combine like terms
Now, we combine the terms on the left side:
\[ t^2 + 2t + 5 + 2t^2 + t - 2 = 2 \]
This simplifies to:
\[ 3t^2 + 3t + 3 = 2 \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to set it to zero:
\[ 3t^2 + 3t + 3 - 2 = 0 \]
This simplifies to:
\[ 3t^2 + 3t + 1 = 0 \]

### Step 4: Use the discriminant to find the nature of the roots
To determine the values of \( t \), we calculate the discriminant \( D \) of the quadratic equation \( at^2 + bt + c = 0 \) where \( a = 3, b = 3, c = 1 \):
\[ D = b^2 - 4ac = 3^2 - 4 \cdot 3 \cdot 1 = 9 - 12 = -3 \]

### Step 5: Analyze the discriminant
Since the discriminant \( D < 0 \), this indicates that the quadratic equation has no real roots. Therefore, there are no real values of \( t \) for which the point lies on the line \( x + y = 2 \).

### Conclusion
The point \( (t^2 + 2t + 5, 2t^2 + t - 2) \) does not lie on the line \( x + y = 2 \) for any real value of \( t \).

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