The nearest point from the origin is

Options

A. (2, -3)
B. (6,0)
C. (-2,-1)
D. (3,5)

Answer

To find the nearest point from the origin (0,0) among the given points, we will calculate the distance from the origin to each point using the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

In our case, we will calculate the distance from the origin (0,0) to each of the following points:
1. \(P_1 = (2, -3)\)
2. \(P_2 = (6, 0)\)
3. \(P_3 = (-2, -1)\)
4. \(P_4 = (3, 5)\)

### Step 1: Calculate the distance from the origin to \(P_1 = (2, -3)\)

Using the distance formula:

\[ d_1 = \sqrt{(2 - 0)^2 + (-3 - 0)^2} = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 2: Calculate the distance from the origin to \(P_2 = (6, 0)\)

Using the distance formula:

\[ d_2 = \sqrt{(6 - 0)^2 + (0 - 0)^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6 \]

### Step 3: Calculate the distance from the origin to \(P_3 = (-2, -1)\)

Using the distance formula:

\[ d_3 = \sqrt{(-2 - 0)^2 + (-1 - 0)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]

### Step 4: Calculate the distance from the origin to \(P_4 = (3, 5)\)

Using the distance formula:

\[ d_4 = \sqrt{(3 - 0)^2 + (5 - 0)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \]

### Step 5: Compare the distances

Now we have the distances:
- \(d_1 = \sqrt{13}\)
- \(d_2 = 6\)
- \(d_3 = \sqrt{5}\)
- \(d_4 = \sqrt{34}\)

To find the nearest point, we compare these distances. The smallest distance is \(d_3 = \sqrt{5}\).

### Conclusion

The nearest point from the origin is \(P_3 = (-2, -1)\).

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