1. What is the equation of the circle with center (-4,-3) that passes through the point (6,2)?
A) (x-4)^2+(y-3)^2=25
B) (x-4)^2+(y-3)^2=125
C) (x-(-4))^2+(y-(-3))^2=25
D) (x-(-4))^2+(y-(-3))^2=25
I'm really not sure how to do this at all, so any help would be absolutely fantastic. Thank you so much.
When I solve the first two, I both get positive (4,3). Which isn't correct. And i'm confused.
in general if the centre is (h,k), the equation is
(x-h)^2 + (y-k)^2 = r^2
notice that the sign inside the bracket is opposite to the sign used in the given point.
so for (-4,-3) it must be
(x-(-4) )^2 + (y - (-3) )^2 = r^2
(x+4)^2 + (y+3)^2 = r^2
to find the r^2, just plug in the given point (6,2)
(6+4)^2 + (2+3)^2 = r^2
125 = r^2
by the looks of the choices , none of them are correct, unless you have a typo in either C or D
(notice they are the same)
Furthermore, "they" should not have left it as
(x-(-4))^2 etc
correct answer:
(x+4)^2 + (y+3)^2 = 125
Oh my apologies! D was 125, not 25. Thank you so much though, I was quiet confused as well since I was thinking they were both 25.
To find the equation of a circle with a given center and passing through a specific point, follow these steps:
Step 1: Identify the center coordinates.
In this case, the center coordinates are given as (-4, -3).
Step 2: Identify the point that the circle passes through.
In this case, the point is (6, 2).
Step 3: Determine the radius of the circle.
The radius is the distance between the center and the point that the circle passes through. You can use the distance formula to find it:
radius = √[(x2 - x1)^2 + (y2 - y1)^2]
Using the given coordinates, we get:
radius = √[(6 - (-4))^2 + (2 - (-3))^2]
radius = √[(6 + 4)^2 + (2 + 3)^2]
radius = √[10^2 + 5^2]
radius = √(100 + 25)
radius = √125
Step 4: Write the equation.
The standard equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center coordinates, and r represents the radius.
Substituting the center coordinates and the radius we found into the equation, we have:
(x - (-4))^2 + (y - (-3))^2 = (√125)^2
(x + 4)^2 + (y + 3)^2 = 125
Therefore, option C is the correct answer:
(x - (-4))^2 + (y - (-3))^2 = 25.
To find the equation of a circle, you need to use the standard form equation:
(x - h)^2 + (y - k)^2 = r^2
Where (h, k) represents the center of the circle and r represents the radius.
In this problem, the center of the circle is given as (-4, -3), and it passes through the point (6, 2).
To find the radius, you need to calculate the distance between the center of the circle and the given point. The distance formula is:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
Plugging in the coordinates, you get:
d = √[(6 - (-4))^2 + (2 - (-3))^2]
= √[(6 + 4)^2 + (2 + 3)^2]
= √[10^2 + 5^2]
= √[100 + 25]
= √125
= 5√5
Now that you have the radius, you can substitute the values into the standard form equation:
(x - (-4))^2 + (y - (-3))^2 = (5√5)^2
(x + 4)^2 + (y + 3)^2 = 25 * 5
(x + 4)^2 + (y + 3)^2 = 125
Therefore, the correct answer is option B: (x - 4)^2 + (y - 3)^2 = 125.