Find the centre and the radius of the circle x2+y2+8x+10y−8=0.
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Step-by-Step Solution
Step 1: Recall the Standard Equation of a Circle
The standard equation of a circle with center (h, k) and radius r is given by (x−h)2+(y−k)2=r2. We need to transform the given equation into this standard form to identify the center and radius.
Step 2: Rearrange and Group Terms
First, rearrange the terms of the given equation, grouping the x terms together and the y terms together, and moving the constant term to the right side of the equation.
Step 3: Complete the Square for x-terms
To complete the square for the x -terms, take half of the coefficient of x (which is 8), square it, and add it to both sides of the equation. Half of 8 is 4, and 42 is 16.
Step 4: Complete the Square for y-terms
Similarly, for the y -terms, take half of the coefficient of y (which is 10), square it, and add it to both sides of the equation. Half of 10 is 5, and 52 is 25.
Step 5: Rewrite in Standard Form
Now, substitute the completed squares back into the equation and sum the constants on the right side. This gives the equation in the standard form (x−h)2+(y−k)2=r2.
Step 6: Identify Center and Radius
By comparing (x+4)2+(y+5)2=49 with the standard form (x−h)2+(y−k)2=r2, we can identify the center (h, k) and the radius r. Here, h=−4, k=−5, and r2=49, so r=49=7.