Question 6, 2.4.23 HW Score: \( 22.22 \%, 2.67 \) of Part 2 of 3 Points: 0 of 1 Write the standard form of the equation and the general form of the equation of the circle of radius r and center ( \( \mathrm{h}, \mathrm{k} \) ). Graph the circle. \[ r=\sqrt{3} ;(h, k)=(4,-2) \] The equation for the circle in standard form is \( (x-4)^{2}+(y+2)^{2}=3 \). (Simplify your answer.) The equation for the circle in general form is \( \square \) . (Simplify your answer.)

To convert the standard form of the circle's equation into the general form, start with the equation \( (x - 4)^{2} + (y + 2)^{2} = 3 \). Expanding this, we have: \[ (x - 4)^{2} = x^2 - 8x + 16 \] \[ (y + 2)^{2} = y^{2} + 4y + 4 \] Combining these yields: \[ x^2 - 8x + 16 + y^{2} + 4y + 4 = 3 \] Now, simplify by moving 3 to the left side: \[ x^2 - 8x + y^2 + 4y + 17 = 0 \] This is the equation in general form. So, the general form of the equation is: \[ x^{2} + y^{2} - 8x + 4y + 17 = 0 \] And there you have it! Your graphical representation of the circle would be centered at the point (4, -2) with a radius of \(\sqrt{3}\), which means the circle would look perfectly round and cozy on your graph! 🎨✨ If you were to plot this circle on a coordinate system, simply locate the center at (4, -2) and then, from this point, measure roughly \(1.73\) units in every direction to sketch the circle’s perimeter. Happy graphing! 📈✍️