Question

1. which of the following points lies on the circle with equation ((x - 4)^2 + (y + 2)^2 = 36)?

a. ((4, 8))
b. ((0, 2))
c. ((4, -2))
d. ((4, 4))

Explanation:

Step1: Recall the circle equation

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius. For the given circle \((x - 4)^2 + (y + 2)^2 = 36\), the center is \((4, -2)\) and radius \(r = 6\) (since \(r^2 = 36\)). To check if a point \((x,y)\) lies on the circle, substitute \(x\) and \(y\) into the equation and see if it equals \(36\).

Step2: Check option a: \((4,8)\)

Substitute \(x = 4\), \(y = 8\) into the equation:
\((4 - 4)^2 + (8 + 2)^2 = 0^2 + 10^2 = 0 + 100 = 100\).
\(100
eq 36\), so \((4,8)\) does not lie on the circle.

Step3: Check option b: \((0,2)\)

Substitute \(x = 0\), \(y = 2\) into the equation:
\((0 - 4)^2 + (2 + 2)^2 = (-4)^2 + 4^2 = 16 + 16 = 32\).
\(32
eq 36\), so \((0,2)\) does not lie on the circle.

Step4: Check option c: \((4,-2)\)

Substitute \(x = 4\), \(y = -2\) into the equation:
\((4 - 4)^2 + (-2 + 2)^2 = 0^2 + 0^2 = 0 + 0 = 0\).
\(0
eq 36\), so \((4,-2)\) is the center, not on the circle.

Step5: Check option d: \((4,4)\)

Substitute \(x = 4\), \(y = 4\) into the equation:
\((4 - 4)^2 + (4 + 2)^2 = 0^2 + 6^2 = 0 + 36 = 36\).
This equals \(36\), so \((4,4)\) lies on the circle. Wait, no—wait, let's recheck. Wait, the center is \((4, -2)\), radius 6. Wait, maybe I made a mistake. Wait, option a: \((4,8)\): \(y\) - coordinate difference from center \((4,-2)\) is \(8 - (-2) = 10\), which is more than radius 6? Wait no, the radius is 6, so the distance from center \((4,-2)\) to \((4,8)\) is \(|8 - (-2)| = 10\), which is more than 6. Wait, option d: \((4,4)\): distance from \((4,-2)\) is \(|4 - (-2)| = 6\), which is equal to the radius. Wait, but let's re - substitute option a again. Wait, the equation is \((x - 4)^2+(y + 2)^2 = 36\). For option a: \((4,8)\): \((4 - 4)^2+(8 + 2)^2=0 + 100 = 100\). Option d: \((4,4)\): \((4 - 4)^2+(4 + 2)^2=0 + 36 = 36\). Wait, but also check option a again. Wait, maybe I misread the options. Wait, the options are a. (4,8), b. (0,2), c. (4,-2), d. (4,4). Wait, but let's check option a again. Wait, no—wait, the correct substitution for option a: \((4,8)\): \(y = 8\), so \(y+2 = 10\), squared is 100. Option d: \(y = 4\), \(y + 2 = 6\), squared is 36. So \((4,4)\) gives 36. But wait, another way: the circle has center \((4, -2)\) and radius 6. So the vertical line \(x = 4\) (since center is on \(x = 4\)) will have points on the circle where \(y=-2\pm6\), so \(y = 4\) (since \(-2 + 6 = 4\)) and \(y=-8\) (since \(-2-6=-8\)). So \((4,4)\) is on the circle (distance from center is 6), and \((4,-8)\) would also be. So among the options, \((4,4)\) (option d) satisfies the equation. Wait, but earlier when I checked option d, I thought it was correct. Wait, but let's re - check all options again carefully.

Wait, option a: \((4,8)\): \((4 - 4)^2+(8 + 2)^2=0 + 100 = 100
eq36\)

Option b: \((0,2)\): \((0 - 4)^2+(2 + 2)^2=16 + 16 = 32
eq36\)

Option c: \((4,-2)\): \((4 - 4)^2+(-2 + 2)^2=0 + 0 = 0
eq36\)

Option d: \((4,4)\): \((4 - 4)^2+(4 + 2)^2=0 + 36 = 36\)

So option d satisfies the equation. Wait, but initially I thought maybe I made a mistake, but the substitution shows that \((4,4)\) gives 36, which matches the right - hand side of the circle equation.

Answer:

d. (4, 4)