Answer:
x² + y² = 49
Explanation:
The circle being centred at the origin simplifies things a lot. Also, given an x-intercept point makes it easy to find the origin.
The equation for a circle is:
(x - x₁)² + (y - y₁)² = r²
Where (x₁, y₁) is the centre point of the circle.
In this case, the centre of the circle is (0, 0), giving us:
x² + y² = r²
We are told that the circle intercepts (-7, 0), which tells us that it has a radius of 7.
We can simply plug that into the given equation and say:
x² + y² = 7²
x² + y² = 49
More:
***NOTE***
This is not part of the answer to the question, just an expansion of it.
Generically speaking, The equation for a circle is:
(x - x₁)² + (y - y₁)² = r²
Which applies when the centre of the circle is not at the origin. For example, if the given circle was centred around the point (6, 8), with the same radius of 7, then the circle's equation would be (x - 6)² + (y - 8)² = 49.
The other complication that could show up is if the given border point does not have the same x or y coordinate as the circle's centre. For example, if we are given a centre point of (-4, 6) and a border point of (-1, 2), we would first need to use the Pythagorean theorem to find the circle's radius:
r² = (x - x₁)² + (y - y₁)²
r² = (-4 - (-1))² + (6 - 2)²
r² = (-3)² + 4²
r² = 9 + 16
r² = 25
r = 5
We can then plug that into the proper equation, giving us:
(x + 4)² + (y - 6)² = 25
You can also expand and group terms to express that in general form:
x² + 8x + 16 + y² - 12y + 36 = 26
x² + y² + 8x - 12y + 26 = 0
Again, note that this part of the answer is not related to the original question, but an expansion on it.