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=0.5x+4 intersect the circle with radius 3 and center (0, 4)?
x=?

y=?

Answer 1

Answer:

At intersection

x=2.68

y=5.34

Step-by-step explanation:

Equation of circle with center (h,k) and radius r is

$(x-h)^{2}+(y-k)^{2}=r^2\\\\With\,\, center\,\,(0,4)\,\, and\,\, r=3\\\\(x-0)^{2}+(y-4)^{2}=3^2\\\\x^2+(y-4)^{2}=9---(1)\\\\Equation\,\,of intersecting\,\,line\\y=0.5x+4---(2)$

To find point of intersection:

From (2)

$x=\frac{y-4}{0.5}=2(y-4)\\\\x^2=4(y-4)^{2}\\\\Put\,\,in\,\,(1)\\\\4(y-4)^{2}+(y-4)^{2}=9\\\\5(y-4)^{2}=9\\\\y-4=\frac{3}{\sqrt{5}}\\\\y=1.34+4\\\\y=5.34\\\\x=2(5.34-4)\\\\x=2(0.34)\\\\x=2.68$

At intersection

x=2.68

y=5.34

This point of intersection can be checked in graph attached below