- Substitute \(x = 4\) into the circle's equation \(x^2 + y^2 - 10x + 6y + 30 = 0\).
- Simplify the resulting equation to the quadratic form \(y^2 + 6y + 6 = 0\).
- Solve for \(y\) using the quadratic formula to obtain the intersection points.
- The final \(y\)-coordinates are \(\boxed{-3 \pm \sqrt{3}}\).
Explanation
- Analyze the problem and given conditions.
The circle \(C\) is defined by the equation \(x^2 + y^2 - 10x + 6y + 30 = 0\). We are interested in the intersection of this circle with the vertical line defined by the equation \(x = 4\). To find the intersection points, we must determine the \(y\)-coordinates that satisfy both the equation of the circle and the equation of the line.
- Substitute the line equation into the circle equation.
Substitute \(x = 4\) into the circle's equation to obtain a quadratic equation solely in terms of \(y\):
\[(4)^2 + y^2 - 10(4) + 6y + 30 = 0\]
Simplifying this:
\[16 + y^2 - 40 + 6y + 30 = 0\]
Grouping the constant terms:
\[y^2 + 6y + (16 - 40 + 30) = 0\]
\[y^2 + 6y + 6 = 0\]
- Solve for \(y\) using the quadratic formula.
We now solve the quadratic equation \(y^2 + 6y + 6 = 0\) for \(y\). Using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=6\), and \(c=6\):
\[y = \frac{-6 \pm \sqrt{6^2 - 4(1)(6)}}{2(1)}\]
\[y = \frac{-6 \pm \sqrt{36 - 24}}{2}\]
\[y = \frac{-6 \pm \sqrt{12}}{2}\]
Since \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\), we have:
\[y = \frac{-6 \pm 2\sqrt{3}}{2}\]
Dividing both terms in the numerator by 2:
\[y = -3 \pm \sqrt{3}\]
Thus, the \(y\)-coordinates are \(-3 + \sqrt{3}\) and \(-3 - \sqrt{3}\).
Examples
This process of finding the intersection of a circle and a line is foundational in coordinate geometry and calculus. For example, if you were modeling a path of a circular fountain in a garden layout and wanted to place a straight stone path across it at a specific position, this math allows you to pinpoint the exact entry and exit coordinates for your construction materials. It's an essential skill for accurate design and architectural planning.