The equation of the circle is x(squared)+y(squared)-3x -4=0.Find :
(i)the coordinates of its center and the radius
(ii) the coordinates of the points at which it cuts the x-axis
x^2 -3 x + y^2 = 4
x^2 -3x + 9/4 + y^2 = 4 + 9/4
(x - 3/2)^2 + y^2 = 25/4
center at (3/2 , 0)
radius = 5/2
when y = 0
(x-3/2) = +/- 5/2
x = 1.5+2.5 = 4 so (4,0)
x = 1.5-2.5 = -1 so (-1,0)
post it.
To find the coordinates of the center and the radius of a circle given its equation, we need to rewrite the equation in the standard form of a circle equation, which is:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center of the circle and r represents its radius.
Let's start by rewriting the given equation:
x^2 + y^2 - 3x - 4 = 0
To complete the square and convert this equation into the standard form, we need to group the x and y terms, and then add a value inside the parentheses on both sides of the equation:
(x^2 - 3x) + (y^2) = 4
Next, we need to add half of the coefficient of x squared (which is -3), squared, to both sides of the equation. Similarly, we add half of the coefficient of y squared (which is 0), squared, to both sides:
(x^2 - 3x + 9/4) + (y^2) = 4 + 9/4
Simplifying further:
(x^2 - 3x + 9/4) + (y^2) = 16/4 + 9/4
(x^2 - 3x + 9/4) + (y^2) = 25/4
Finally, we can rewrite this as:
(x - 3/2)^2 + (y - 0)^2 = (5/2)^2
Comparing this equation to the standard form, we can see that:
Center = (h, k) = (3/2, 0)
Radius = r = 5/2
(i) The coordinates of the center are (3/2, 0), and the radius is 5/2.
(ii) To find the points at which the circle intersects the x-axis, we can substitute y = 0 into the equation. This will give us a quadratic equation in x, and solving it will give us the x-coordinates of the intersection points.
Substituting y = 0 into the equation, we have:
(x - 3/2)^2 + (0 - 0)^2 = (5/2)^2
(x - 3/2)^2 = 25/4
Take the square root of both sides:
x - 3/2 = ±(5/2)
Simplifying:
x = 3/2 ± (5/2)
Thus, the coordinates of the points at which the circle intersects the x-axis are:
(x1, 0) = (3/2 + 5/2, 0) = (4, 0)
(x2, 0) = (3/2 - 5/2, 0) = (-1, 0)
Therefore, the circle intersects the x-axis at the points (4, 0) and (-1, 0).