Understanding the Parabola and Tangent Line
To determine if the line y = c is a tangent to a parabola, we can analyze the equation of the parabola given as y = (7/2)x^2.
Finding the Tangent Condition
A line is tangent to a curve if they intersect at exactly one point. For a line y = c to be tangent to the parabola, the following condition must hold:
- Set (7/2)x^2 = c to find the x-coordinates where they intersect.
Setting Up the Equation
This results in the quadratic equation:
- (7/2)x^2 - c = 0
For this equation to have exactly one solution (indicating tangency), the discriminant must be zero. The general form of a quadratic equation is ax^2 + bx + c = 0, where the discriminant D is given by:
- D = b^2 - 4ac
Here, a = 7/2, b = 0, and c = -c. Thus, the discriminant becomes:
- D = 0^2 - 4*(7/2)*(-c) = 14c
Condition for Tangency
Setting the discriminant to zero gives us:
- 14c = 0
This implies:
- c = 0
Conclusion
Since the only value for c that satisfies the tangency condition is 0, the correct answer is indeed:
- d) none of these (since 0 is not listed in the options provided).
This thorough analysis shows that the line y = c can only be tangent to the parabola when c equals 0, indicating that the correct answer is option 'D'.