Explanation
Let the equation of the general form of the required circle be
x2+y2+2gx+2fy+c=0................(1)
According to the problem, the above equation of the circle passes through the points (0,6),(0,0) and (8,0). Therefore,
36+12f+c=0 ………. (2)
c=0 ……………. (3)
64+16g+c=0 ……………. (4)
Putting c=0 in (2), we obtain f=−3. Similarly put c=0 in (4), we obtain g=−4
Substituting the values of g,f and c in (1), we obtain the equation of the required circle as:
x2+y2+2(−4)x+2(−2)y+0=0 that is
x2+y2−8x−4y+0=0 can be rewritten as
x2+y2−8x−4y+16+9=0+16+9
(x−4)2+(y−3)2=25
Therefore, the equation of circle is (x−4)2+(y−3)2=25.
Let s: x^2 + y^2 + 2gx + 2fy +c = 0 be the equation of the circle
S(0,2) = 0
\implies 4 + 4f + c =0 -eq.1
Intercept on x-axis is given by
2\sqrt{g^2 - c} = 4
g^2 – c = 4 -eq.2
Since x = 0 is a tangent
Radius = perpendicular distance from the center to tangent
$$\sqrt {g^2 + f^2 - c} = \dfrac{|-g|}{\sqrt{1 + 0}$$
g^2 + f^2 - c = g^2
f^2 = c
Substituting in eq.1
4 + 4f + f^2 =0 \implies (f+2)^2 = 0 \implies f = -2
\implies c = 4
From eq.2
g^2 – 4 = 4
g = \pm 2 \sqrt{2}
Equation of circle is
x^2 +y^2 \pm 4\sqrt 2 x – 4y + 4 = 0
\implies x^2 +y^2 – 4(\sqrt 2 x + y) + 4 = 0
The circumference of the circle is given as 10\pi .
C = 2\pi r
10\pi = 2\pi r
r = \frac{{10\pi }}{{2\pi }}
r = 5
The general form of the equation is,
{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}
The center of the circle is given as \left( {h,k} \right) = \left( {2, - 3} \right)
{\left( {x - 2} \right)^2} + {\left( {y - \left( { - 3} \right)} \right)^2} = {\left( 5 \right)^2}
{\left( {x - 2} \right)^2} + {\left( {y + 3} \right)^2} = 25
{x^2} + 4 - 4x + {y^2} + 9 + 6y = 25
{x^2} + {y^2} - 4x + 6y + 13 = 25
{x^2} + {y^2} - 4x + 6y - 25 + 13 = 0
{x^2} + {y^2} - 4x + 6y - 12 = 0
Therefore, the equation of the circle is,
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