Find all points on the $$ x $$-axis that are 12 units from the point $$ (4,-6) $$

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UpStudy AI · Accepted Answer

To find all points on the $$ x $$-axis that are 12 units away from the point $$ (4, -6) $$, follow these steps:

1. **Represent the Points on the $$ x $$-axis:**
   Points on the $$ x $$-axis have the form $$ (x, 0) $$.

2. **Use the Distance Formula:**
   The distance between $$ (x, 0) $$ and $$ (4, -6) $$ is given by:
   $$
   \sqrt{(x - 4)^2 + (0 - (-6))^2} = 12
   $$

3. **Simplify the Equation:**
   $$
   \sqrt{(x - 4)^2 + 36} = 12
   $$
   Square both sides to eliminate the square root:
   $$
   (x - 4)^2 + 36 = 144
   $$
   $$
   (x - 4)^2 = 108
   $$
   $$
   x - 4 = \pm \sqrt{108}
   $$
   
4. **Simplify the Square Root:**
   $$
   \sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3}
   $$
   
5. **Solve for $$ x $$:**
   $$
   x = 4 \pm 6\sqrt{3}
   $$

6. **Write the Points:**
   The two points on the $$ x $$-axis are:
   $$
   \left(4 + 6\sqrt{3},\ 0\right) \quad \text{and} \quad \left(4 - 6\sqrt{3},\ 0\right)
   $$

**Final Answer:**
All such points are (4 + 6√3, 0) and (4 – 6√3, 0).
The points on the $$ x $$-axis that are 12 units from $$ (4, -6) $$ are $$ (4 + 6\sqrt{3}, 0) $$ and $$ (4 - 6\sqrt{3}, 0) $$.

Find all points on the -axis that are 12 units from the point

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