The minimum distance of a point on the curve `y=x^2-4` from origin ,

Options

A. `sqrt(5)/2`
B. `sqrt(19)/2`
C. `sqrt(15/2)`
D. `sqrt(19/2)`

Explanation

To find the minimum distance of a point on the curve \( y = x^2 - 4 \) from the origin, we can follow these steps:

### Step 1: Define the point on the curve
Let the point on the curve be \( (t, t^2 - 4) \), where \( t \) is a variable representing the x-coordinate.

### Step 2: Calculate the distance from the origin
The distance \( S \) from the origin \( (0, 0) \) to the point \( (t, t^2 - 4) \) is given by the formula:
\[ S = \sqrt{t^2 + (t^2 - 4)^2} \]

### Step 3: Simplify the expression for distance
To simplify the calculations, we can minimize \( S^2 \) instead of \( S \):
\[ S^2 = t^2 + (t^2 - 4)^2 \]
Expanding \( (t^2 - 4)^2 \):
\[ (t^2 - 4)^2 = t^4 - 8t^2 + 16 \]
Thus,
\[ S^2 = t^2 + t^4 - 8t^2 + 16 = t^4 - 7t^2 + 16 \]

### Step 4: Find the derivative
To find the minimum, we take the derivative of \( S^2 \) with respect to \( t \):
\[ \frac{d(S^2)}{dt} = 4t^3 - 14t \]

### Step 5: Set the derivative to zero
Setting the derivative equal to zero to find critical points:
\[ 4t^3 - 14t = 0 \]
Factoring out \( 2t \):
\[ 2t(2t^2 - 7) = 0 \]
This gives us:
\[ t = 0 \quad \text{or} \quad 2t^2 - 7 = 0 \Rightarrow t^2 = \frac{7}{2} \Rightarrow t = \pm \sqrt{\frac{7}{2}} \]

### Step 6: Determine the nature of critical points
To determine whether these points are minima or maxima, we calculate the second derivative:
\[ \frac{d^2(S^2)}{dt^2} = 12t^2 - 14 \]
Evaluating at \( t = 0 \):
\[ \frac{d^2(S^2)}{dt^2} \bigg|_{t=0} = -14 \quad (\text{which is negative, indicating a maximum}) \]
Evaluating at \( t = \sqrt{\frac{7}{2}} \):
\[ \frac{d^2(S^2)}{dt^2} \bigg|_{t=\sqrt{\frac{7}{2}}} = 12\left(\frac{7}{2}\right) - 14 = 42 - 14 = 28 \quad (\text{which is positive, indicating a minimum}) \]

### Step 7: Calculate the minimum distance
Now we substitute \( t = \sqrt{\frac{7}{2}} \) into the distance formula:
\[ S^2 = \left(\sqrt{\frac{7}{2}}\right)^4 - 7\left(\sqrt{\frac{7}{2}}\right)^2 + 16 \]
Calculating each term:
\[ \left(\sqrt{\frac{7}{2}}\right)^4 = \left(\frac{7}{2}\right)^2 = \frac{49}{4} \]
\[ -7\left(\sqrt{\frac{7}{2}}\right)^2 = -7 \cdot \frac{7}{2} = -\frac{49}{2} \]
Combining these:
\[ S^2 = \frac{49}{4} - \frac{98}{4} + \frac{64}{4} = \frac{49 - 98 + 64}{4} = \frac{15}{4} \]
Thus,
\[ S = \sqrt{\frac{15}{4}} = \frac{\sqrt{15}}{2} \]

### Final Answer
The minimum distance of a point on the curve \( y = x^2 - 4 \) from the origin is \( \frac{\sqrt{15}}{2} \).