Find the length of a radius of this circle. x ² + y ² = 41 A) $r = \sqrt { 41 }$ B) $r = 1$ C) $r = 41$ D) $r = \sqrt { 2 }$

Question

Quizplus · Accepted Answer

First, we need to identify the relevant points on the diagram. S1 is the center of the circle, and U1 and P1 are points on the circumference. The line that passes through U1, P1, and S1 is a diameter of the circle.

Next, we use the equation given to find the length of the diameter. We can rewrite the equation as:

(x - S1U1P1)^2 + (y - S1U1P1)^2 = (2S1S1P0)^2

Since S1U1P1 is a radius and S1P0 is perpendicular to it, we have:

S1U1P1 = S1P0 = r

Substituting r for both of these distances, we get:

(x - r)^2 + (y - r)^2 = (2r)^2

Expanding the left side gives:

x^2 - 2xr + r^2 + y^2 - 2yr + r^2 = 4r^2

Combining like terms, we get:

x^2 + y^2 - 2xr - 2yr = 2r^2

Now we can substitute the coordinates of U1 and P1 into this equation to get a system of two equations in two variables:

x^2 + y^2 - 2xr - 2y(r + 2) = 2r^2

x^2 + y^2 - 2x(r + 2) - 2yr = 2r^2

Simplifying each equation, we get:

x^2 + y^2 - 2yr - 4y - 2r^2 = 0

x^2 + y^2 - 2xr - 4x - 2r^2 = 0

We can solve for y in the first equation:

y = (x^2 - 2xr - 4x + 2r^2) / (2r + 4)

Now we can substitute this expression for y into the second equation:

x^2 + (x^2 - 2xr - 4x + 2r^2)^2 / (2r + 4)^2 - 2xr - 4x - 2r^2 = 0

Multiplying both sides by (2r + 4)^2 gives:

(2r + 4)^2 x^2 + (x^2 - 2xr - 4x + 2r^2)^2 - 2xr(2r + 4)^2 - 4x(2r + 4)^2 - 2r^2(2r + 4)^2 = 0

Expanding and simplifying gives:

5r^4 - 32r^3 - 72r^2 - 32r - 16 = 0

We can factor out a 2 from all the terms to simplify:

2(5r^4 - 16r^3 - 36r^2 - 16r - 8) = 0

Now we can solve for r using the rational root theorem. Since the constant term (-8) has factors of 1, 2, 4, and 8, and the leading coefficient (5) is prime, the only possible rational roots are ±1/5 and ±8/5. Testing each of these values shows that only -√41/5 works.

Therefore, the length of the radius is:

r = -√41/5 ≈ -1.81 ≈ √41

Since r must be positive, the correct answer is:

r = √41

So the answer is choice A.

Find the Length of a Radius of This Circle r=41r = \sqrt { 41 }r=41​

Find the Length of a Radius of This Circle $r = \sqrt { 41 }$