To determine the equation of the line that is tangent to the circle [tex]\( x^2 + y^2 = 13 \)[/tex] at the point [tex]\( (2, 3) \)[/tex], we can follow these steps:
1. Understand the Circle's Equation:
The general equation of the circle is [tex]\( x^2 + y^2 = r^2 \)[/tex]. Here, [tex]\( r^2 \)[/tex] is given as 13. Therefore, the equation of the circle is [tex]\( x^2 + y^2 = 13 \)[/tex].
2. Identify the Point of Tangency:
We need to find the tangent line at the point [tex]\( (2, 3) \)[/tex].
3. Formula for the Tangent Line:
For a circle centered at the origin with radius [tex]\( r \)[/tex], the tangent line at any point [tex]\( (x_0, y_0) \)[/tex] on the circle can be found using the formula:
[tex]\[
x_0 \cdot x + y_0 \cdot y = r^2
\][/tex]
4. Substitute the Point (2,3) into the Formula:
Given that the point of tangency is [tex]\( (2, 3) \)[/tex], we substitute [tex]\( x_0 = 2 \)[/tex] and [tex]\( y_0 = 3 \)[/tex] into the formula above:
[tex]\[
2 \cdot x + 3 \cdot y = r^2
\][/tex]
5. Substitute the Value of [tex]\( r^2 \)[/tex]:
We know from the circle’s equation that [tex]\( r^2 = 13 \)[/tex]. So the equation becomes:
[tex]\[
2x + 3y = 13
\][/tex]
Thus, the equation of the line that is tangent to the circle [tex]\( x^2 + y^2 = 13 \)[/tex] at the point [tex]\( (2, 3) \)[/tex] is:
[tex]\[
2x + 3y = 13
\][/tex]
This concludes our step-by-step solution to finding the equation of the tangent line.
