Question

The endpoints of the longest chord on a circle are (4, 5.5) and (4, 10.5).the center of the circle is at the point , and its radius is units. the equation of this circle in standard form is .

Answer 1

 The midpoint of the points ( 4, 5.5 ) and ( 4, 10.5 ) is the center of the circle:
 C =$(\frac{4+4}{2}, \frac{5.5+10.5}{2})$
 C = ( 4, 8 ),  h=4, k=8
 r =$\sqrt{(4-4)^{2} +(8-5.5)^{2} } = \sqrt{0+2.5^{2} }$
 r = 2.5
 The equation of the circle in standard form:
 ( x - h )² + ( y - k )² = r²
 ( x - 4 )² + ( y - 8 ) = 6.25    

Answer 2

Based on the calculations, the equation of this circle in standard form is equal to (x - 4)² + (y - 8)² = 5²

The equation of a circle.

Mathematically, the standard form of the equation of a circle is given by;

(x - h)² + (y - k)² = r²

Where:

• h and k represents the coordinates at the center.

• r is the radius of a circle.

The midpoint of the given points represent the center of this circle:

h = (4 + 4)/2 = 4

k = (5.5 + 10.5)/2 = 8

Next, we would determine the radius by using the distance formula for coordinates:

r = √[(x₂ - x₁)² + (y₂ - y₁)²]

r = √[(4 - 4)² + (10.5 - 5.5)²]

r = √[0² + 5²]

r = √25

r = 5 units.

Therefore, the equation of this circle in standard form is equal to (x - 4)² + (y - 8)² = 5².

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