Question

Ross is analyzing a circle, y2 + x2 = 64, and a linear function g(x). Will they intersect?

Answer 1

The equation of the circle is written in standard form. We know that the center is at (0,0) and the radius is 8.
g(x) is a linear function that passes through the point (0,4). Since this point on the line lies within the circle, we know that the line must intersect the circle at some point.
The answer is yes, they will intersect.
If you have to find the function for g(x):
Slope = m = (-3.5 + 4) / 1 = 1/2
y-intercept = b = -4, since we're given the point (0,-4)
g(x) = (1/2)x - 4

Answer 2

Yes, they will intercept.

The circle, y² + x² = 64, intersects with g(x) at the points (8,0) and (-4.8,-6.4)

Explanation:

x g(x)

−1 −4.5

0 −4

1 −3.5

y2 + x2 = 64

First of, we find the equation for g(x).

It is stated that g(x) is a linear function, hence, it has the equation of straight line.

y = mx + c [where y = g(x), m = slope, c = y-intercept]

Putting the values of x and corresponding y into the equation of a straight line, we can obtain the equation of the straight line.

x y

−1 −4.5

0 −4

1 −3.5

-4.5 = -m + c (from the first line)

-4 = c (from the second line)

-3.5 = m + c (from the third line)

So, from these relation, it is evident that

c = -4, m = 0.5

g(x) = y = 0.5x - 4

The equation of the circle is

y² + x² = 64

If the circle and the line, g(x), are to intersect, they will have the same coordinates At the point of intersection.

Equation of line

y = 0.5x - 4

Substituting this value for y in the equation of the circle

y² + x² = 64

(0.5x - 4)² + x² = 64

0.25x² - 4x + 16 + x² = 64

1.25x² - 4x - 48 = 0

Solvingthe quadratic equation

x = 8 or x = - 4.8

when x = 8, y = 0.5(8) - 4 = 0

when x = 4.8, y = 0.5(-4.8) - 4 = -6.4

therefore, it is evident that the circle, y² + x² = 64, intersects with g(x) At the points (8,0) and (-4.8,-6.4)

Hope this Helps!!+