$$ x^{2}+y^{2}=16 $$ Example 2 Write the equation(s) or coordinates and sketch each locus. (a) The locus of points that are 3 units from $$ y=4 $$.

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To find the locus of points that are 3 units from the line $$ y = 4 $$, we need to consider the distance between a point on the line and the line itself.

The distance between a point $$(x, y)$$ and the line $$y = 4$$ can be calculated using the formula for the distance between a point and a line:

$$ \text{Distance} = \frac{|y - 4|}{\sqrt{1^2 + 0^2}} $$

Since the line is horizontal, the distance formula simplifies to:

$$ \text{Distance} = |y - 4| $$

We want to find the locus of points where this distance is equal to 3 units. Therefore, we set up the equation:

$$ |y - 4| = 3 $$

Solving this equation will give us the coordinates of the points that are 3 units from the line $$ y = 4 $$.

Let's solve the equation $$ |y - 4| = 3 $$ to find the coordinates of the points on the locus.
Solve the equation by following steps:
- step0: Solve for $$y$$:
  $$\left|y-4\right|=3$$
- step1: Separate into possible cases:
  $$\begin{align}&y-4=3\&y-4=-3\end{align}$$
- step2: Solve the equation:
  $$\begin{align}&y=7\&y=1\end{align}$$
- step3: Rewrite:
  $$y_{1}=1,y_{2}=7$$
The solutions to the equation $$ |y - 4| = 3 $$ are $$ y = 1 $$ and $$ y = 7 $$.

Therefore, the locus of points that are 3 units from the line $$ y = 4 $$ consists of two horizontal lines at $$ y = 1 $$ and $$ y = 7 $$.

The coordinates of the points on the locus are $$(x, 1)$$ and $$(x, 7)$$, where $$x$$ can be any real number.
The locus of points 3 units from $$ y = 4 $$ is the horizontal lines $$ y = 1 $$ and $$ y = 7 $$.

\[ x^{2}+y^{2}=16 \] Example 2 Write the equation(s) or coordinates and sketch each locus. (a) The locus of points that are 3 units from \( y=4 \).

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