Question

recall the equation for a circle with center $(h,k)$ and radius $r$. at what point in the first quadrant does the line with equation $y = 2x + 5$ intersect the circle with radius 4 and center $(0,5)$?
$x=$
$y=$
enter your answer correct to 3 decimal places.
question help: video

Explanation:

Step1: Write the equation of the circle

The equation of a circle with center \((h,k)=(0,5)\) and radius \(r = 4\) is \((x - 0)^2+(y - 5)^2=4^2\), which simplifies to \(x^{2}+(y - 5)^{2}=16\).

Step2: Substitute \(y=2x + 5\) into the circle's equation

Substitute \(y\) in the circle's equation: \(x^{2}+(2x + 5-5)^{2}=16\).
Simplify the equation: \(x^{2}+(2x)^{2}=16\), \(x^{2}+4x^{2}=16\), \(5x^{2}=16\), \(x^{2}=\frac{16}{5}\), \(x=\pm\frac{4}{\sqrt{5}}\).
Since the point is in the first quadrant, \(x=\frac{4}{\sqrt{5}}\approx1.789\).

Step3: Find the value of \(y\)

Substitute \(x=\frac{4}{\sqrt{5}}\) into \(y = 2x+5\).
\(y=2\times\frac{4}{\sqrt{5}}+5=\frac{8}{\sqrt{5}}+5\approx3.578 + 5=8.578\).

Answer:

\(x = 1.789\)
\(y = 8.578\)