If each `a_i > 0`, then the shortest distance between the points `(0,-3)` and the curve `y=1+a_1x^2+a_2 x^4+.......+a_n x^(2n)` is

Answer

To find the shortest distance between the point (0, -3) and the curve given by the equation \( y = 1 + a_1 x^2 + a_2 x^4 + \ldots + a_n x^{2n} \), we can follow these steps:

### Step 1: Evaluate the curve at \( x = 0 \)
The first step is to find the value of \( y \) when \( x = 0 \).

\[ y(0) = 1 + a_1(0)^2 + a_2(0)^4 + \ldots + a_n(0)^{2n} = 1 \]

### Step 2: Analyze the curve
The function \( y = 1 + a_1 x^2 + a_2 x^4 + \ldots + a_n x^{2n} \) consists of only even powers of \( x \), which means it is symmetric about the y-axis.

### Step 3: Determine the minimum value of the curve
Since the coefficients \( a_i > 0 \), the curve opens upwards. The minimum value occurs at \( x = 0 \):

\[ \text{Minimum value of } y = 1 \]

### Step 4: Calculate the distance from the point (0, -3) to the point (0, 1)
The distance \( d \) between the point (0, -3) and the point (0, 1) on the curve can be calculated using the distance formula:

\[ d = |y_2 - y_1| = |1 - (-3)| = |1 + 3| = 4 \]

### Conclusion
Thus, the shortest distance between the point (0, -3) and the curve is:

\[ \text{Shortest distance} = 4 \text{ units} \]

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