To determine the reflection of the point [tex]\(\left(2, -3\right)\)[/tex] in the line [tex]\( y = -3x \)[/tex], we need to follow several steps involving the general formula for reflection across a line of the form [tex]\( ax + by + c = 0 \)[/tex].
1. Identify the line equation in standard form:
The line [tex]\( y = -3x \)[/tex] can be rewritten in the standard form [tex]\( ax + by + c = 0 \)[/tex]:
[tex]\[
y + 3x = 0
\][/tex]
Here, [tex]\( a = 3 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = 0 \)[/tex].
2. General formula for reflection:
If [tex]\( (x_1, y_1) \)[/tex] is the original point and [tex]\( (x_2, y_2) \)[/tex] is the reflected point, the reflection formulas are:
[tex]\[
x_2 = x_1 - \frac{2a(ax_1 + by_1 + c)}{a^2 + b^2}
\][/tex]
[tex]\[
y_2 = y_1 - \frac{2b(ax_1 + by_1 + c)}{a^2 + b^2}
\][/tex]
3. Calculate each part:
First, calculate the denominator, which is [tex]\( a^2 + b^2 \)[/tex]:
[tex]\[
a^2 + b^2 = 3^2 + 1^2 = 9 + 1 = 10
\][/tex]
Next, calculate [tex]\( ax_1 + by_1 + c \)[/tex]:
[tex]\[
ax_1 + by_1 + c = 3(2) + 1(-3) + 0 = 6 - 3 = 3
\][/tex]
4. Substitute into the formulas for [tex]\( x_2 \)[/tex] and [tex]\( y_2 \)[/tex]:
[tex]\[
x_2 = 2 - \frac{2 \cdot 3 \cdot 3}{10} = 2 - \frac{18}{10} = 2 - 1.8 = 0.2
\][/tex]
[tex]\[
y_2 = -3 - \frac{2 \cdot 1 \cdot 3}{10} = -3 - \frac{6}{10} = -3 - 0.6 = -3.6
\][/tex]
5. Transform the result back to fraction form:
[tex]\[
0.2 = \frac{1}{5}
\][/tex]
[tex]\[
-3.6 = -\frac{18}{5}
\][/tex]
6. Conclusion:
The reflected point [tex]\( T\left[\begin{array}{c}2 \\ -3\end{array}\right] \)[/tex] is given by:
[tex]\[
T\left[\begin{array}{c}2 \\ -3\end{array}\right] = \left[\begin{array}{c}0.2 \\ -3.6\end{array}\right] = \left[\begin{array}{c}\frac{1}{5} \\ -\frac{18}{5}\end{array}\right]
\][/tex]
Hence, the correct answer from the given options is:
[tex]\[
(i) \left[\begin{array}{c}1 / 5 \\ -18 / 5\end{array}\right]
\][/tex]
