Recall the equation for a circle with center ( h , k ) and radius r

Question

Recall the equation for a circle with center  and radius . At what point in the first quadrant does the line with equation  intersect the circle with radius 3 and center (0, 5)?

X= ?

y= ?

Answer 1

The equation of the circle with center (h, k) and radius r is:

(x-h)² +(y-k)²=r²

If the circle has a center at (0, 5) and radius 3 the above equation becomes:

x² +(y-5)²=3²

We want to know the point where this circle intersects the line y=x+5 in the first quadrant. In order to do that, we substitute y=x+5 into the equation for the circle or:

x² +(x+5-5)²=3²

x² +x² =9

2x² =9

x = 3 sqrt(2)/2 =2.1213

Substituting to this value into the second equation for y, we find:

y = 2.1213 + 5 = 7.1213

Consequently, the line intersects the circle at (2.1213, 7.1213)