Explanation
centre radius form
let (h,k) be the centre of a cicle and r be the radius
$${(x-h)}^{2}+{(y-k)}^{2}={r}^{2}$$
and we talk about centre at origin
than $$(h,k)$$ will become $$(0,0)$$
so, equation will become $${x}^{2}+{y}^{2}={r}^{2}$$
therefore option A will be answer.
Consider the given circle equation.
$${{x}^{2}}+{{y}^{2}}-4x-6y+4=0$$ …… (1)
We know that the general equation of circle,
$${{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0$$ …… (2)
On comparing equation (1) and (2), we get
$$ 2g=-4\Rightarrow g=-2 $$
$$ 2f=-6\Rightarrow f=-3 $$
$$ c=4 $$
So, the centre of the circle
$$ C=\left( -g,-f \right) $$
$$ C=\left( 2,3 \right) $$
We know that the radius of circle,
$$r=\sqrt{{{g}^{2}}+{{f}^{2}}-c}$$
$$ r=\sqrt{{{2}^{2}}+{{3}^{2}}-4} $$
$$ r=\sqrt{4+9-4} $$
$$ r=\sqrt{9}=3 $$
So, the diameter of this circle
$$d=2r=6$$
Hence, this is the answer.
We know that the general equation of parabola is
$$ {{y}^{2}}=4ax $$
$$ {{y}^{2}}=-4ax $$
$$ {{x}^{2}}=4ax $$
$$ {{x}^{2}}=-4ax $$
From option (a),
$${{\left( x-y \right)}^{3}}=3$$
It is not represent the parabola.
From option (b),
$$ \dfrac{x}{y}-\dfrac{y}{x}=0 $$
$$ {{x}^{2}}={{y}^{2}} $$
From option (c),
$$ \dfrac{x}{y}+\dfrac{4}{x}=0 $$
$$ {{x}^{2}}+4y=0 $$
$$ {{x}^{2}}=-4y $$
So, this is represented the parabola.
From option (d),
$${{\left( x+y \right)}^{3}}+3=0$$
Hence, option c represents the parabola.
Let $$\left( {x,y} \right)$$ be point then ATQ
$$\sqrt {{{\left( {x - 2} \right)}^2} + {{\left( {y - 3} \right)}^2} = 5} $$
$${\left( {x - 2} \right)^2} + {\left( {y - 3} \right)^2} = 25$$
$${x^2} + {y^2} + 4 + 9 - 4x - 6y = 25$$
$$ \Rightarrow {x^2} + {y^2} - 4x - 6y - 12 = 0$$
To find circumcentre we write the equation of perpendicular bisectors of two sides and find their intersection,
3x-y-3=0 and 6x+8y-21=0
Their intersection point is ($$\dfrac{3}{2}$$,$$\dfrac{3}{2}$$)
Radius of circumcircle = Distance of ($$\dfrac{3}{2}$$,$$\dfrac{3}{2}$$)
from (2,-2) or any other vertex = $$\dfrac{5}{\sqrt{2}}$$
So equation of circle = ($$x-\frac{3}{2})^2$$ +($$y-\frac{3}{2})^2$$=$$\dfrac{25}{2}$$ which corresponds to B option .
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