the centre of a circle is (2x-1,3x+1). find the x if the circle is passes thuthe centre of a circle is (2x-1,3x+1). find the x if the circle is passes through (-3, -1) and the length of its diameter is 20 units.gh (ro-3, -1) and the length of its diameter is 20 units.
There sure is a lot of gobbledegook there.
If we say that the center lies on the line
x = 2t-1
y = 3t+1
Apparently the center of the circle lies on the line y = (3x+5)/2
The circle passes through (-3,-1)
So, we need
(-3-h)^2 + (-1-(3h+5)/2)^2 = 100
h=3 or -105/13
so, k= 7 or -125/13
h=3 means x = 2
h = -105/13 means x = -223/13
So, there are two circles that meet the need,with centers at
(3,7) or (-105/13,-125/13)
The graph below shows that the circle intersects the line at (-3,-1)
http://www.wolframalpha.com/input/?i=plot+(x-3)%5E2+%2B+(y-7)%5E2+%3D+100,+y%3D(3x%2B5)%2F2
To find the value of x, we can use the information provided about the center of the circle, the point it passes through, and the length of its diameter.
The center of the circle is given as (2x-1, 3x+1). We can use this to find the coordinates of the center of the circle by equating them to the given coordinates:
2x-1 = -3 and 3x+1 = -1
Solving these equations, we get:
2x = -2 and 3x = -2
Dividing both sides by 2 and 3 respectively, we find:
x = -1 and x = -2/3
So, we have two possible values for x: -1 and -2/3.
Now, let's use the fact that the circle passes through the point (-3, -1) and the length of its diameter is 20 units.
The distance between the center of the circle and the point (-3, -1) is the radius of the circle.
Using the distance formula, the radius of the circle can be calculated as:
sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the values, we get:
sqrt((-3 - (2x-1))^2 + (-1 - (3x+1))^2) = 20
Simplifying this equation, we get:
sqrt((-3 - 2x + 1)^2 + (-1 - 3x - 1)^2) = 20
Simplifying further, we have:
sqrt((4 - 2x)^2 + (-2 - 3x)^2) = 20
Squaring both sides of the equation, we get:
(4 - 2x)^2 + (-2 - 3x)^2 = 400
Expanding and simplifying this equation, we have:
4 - 16x + 4x^2 + 4 + 12x + 9x^2 = 400
Combining like terms and rearranging the equation, we get:
13x^2 - 4x - 392 = 0
Now we have a quadratic equation. We can solve this equation by factoring, completing the square, or using the quadratic formula.
Solving this equation, we find the two possible values for x:
x = 4/13 and x = -14/13
Therefore, the two possible values of x that satisfy the given conditions are -1, -2/3, 4/13, and -14/13.