# A circle has a center that falls on the line #y = 2x +7 # and passes through #(4 ,4 )# and #(1 ,2 )#. What is the equation of the circle?

The standard Cartesian form for the equation of a circle is:

Use equation [1] and the two points to write two equations:

The third equation is written by substituting the center-point to the given line:

Expand equations [2] and [3]:

Subtract equation [6] from equation [5]:

Use equation [4] to substitute into equation [7]:

Use equation [4] to find the value of k:

Use equation [3] to find the value of r:

Use equation [1] to write the equation of the circle:

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The equation of the circle with a center that falls on the line ( y = 2x + 7 ) and passes through the points (4, 4) and (1, 2) is:

[ (x - h)^2 + (y - k)^2 = r^2 ]

Where ( (h, k) ) represents the center of the circle and ( r ) is the radius.

First, we need to find the center of the circle. Since the center falls on the line ( y = 2x + 7 ), we substitute this equation into the general form of the circle equation:

[ (x - h)^2 + (2x + 7 - k)^2 = r^2 ]

Next, we use the given points (4, 4) and (1, 2) to solve for ( h ), ( k ), and ( r ). Plugging in the coordinates of each point into the circle equation, we can form two equations:

[ (4 - h)^2 + (2(4) + 7 - k)^2 = r^2 ] [ (1 - h)^2 + (2(1) + 7 - k)^2 = r^2 ]

By solving these equations simultaneously, we can find the values of ( h ), ( k ), and ( r ). Substituting the values of ( h ) and ( k ) into the equation of the circle will give us the final equation.

This process involves algebraic manipulation and solving equations to find the center and radius of the circle.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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