Find the position of the following points with respect to the parabola `y^(2)=16x`
(i) (4,-8) , (ii) (2,4)
(iii) (0,1) , (iv) (-2,8)

Answer

To find the position of the given points with respect to the parabola defined by the equation \( y^2 = 16x \), we can follow these steps:

### Step 1: Identify the Parabola's Parameters
The standard form of the parabola is \( y^2 = 4ax \). Here, \( 4a = 16 \), which gives \( a = 4 \).

### Step 2: Determine the Conditions
For a point \( (x_1, y_1) \):
- If \( y_1^2 > 4ax_1 \), the point is **outside** the parabola.
- If \( y_1^2 < 4ax_1 \), the point is **inside** the parabola.
- If \( y_1^2 = 4ax_1 \), the point is **on** the parabola.

### Step 3: Analyze Each Point

#### (i) Point (4, -8)
1. Calculate \( y_1^2 \):
\[ y_1^2 = (-8)^2 = 64 \]
2. Calculate \( 4ax_1 \):
\[ 4ax_1 = 16 \times 4 = 64 \]
3. Compare:
\[ 64 = 64 \quad \text{(on the parabola)} \]

#### (ii) Point (2, 4)
1. Calculate \( y_1^2 \):
\[ y_1^2 = 4^2 = 16 \]
2. Calculate \( 4ax_1 \):
\[ 4ax_1 = 16 \times 2 = 32 \]
3. Compare:
\[ 16 < 32 \quad \text{(inside the parabola)} \]

#### (iii) Point (0, 1)
1. Calculate \( y_1^2 \):
\[ y_1^2 = 1^2 = 1 \]
2. Calculate \( 4ax_1 \):
\[ 4ax_1 = 16 \times 0 = 0 \]
3. Compare:
\[ 1 > 0 \quad \text{(outside the parabola)} \]

#### (iv) Point (-2, 8)
1. Calculate \( y_1^2 \):
\[ y_1^2 = 8^2 = 64 \]
2. Calculate \( 4ax_1 \):
\[ 4ax_1 = 16 \times (-2) = -32 \]
3. Compare:
\[ 64 > -32 \quad \text{(outside the parabola)} \]

### Summary of Results
- Point (4, -8): **On the parabola**
- Point (2, 4): **Inside the parabola**
- Point (0, 1): **Outside the parabola**
- Point (-2, 8): **Outside the parabola**