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In this lesson, we will learn how to derive a circle equation given its center and radius and how to extract the radius and center of a circle from its equation.

Q1:

The equation ( π₯ β 3 ) + ( π¦ + 2 ) = 1 0 0 2 2 describes a circle. Find its center and radius.

Q2:

By completing the square, find the centre and radius of the circle π₯ β 4 π₯ + π¦ β 4 π¦ β 8 = 0 2 2 .

Q3:

A circle is the translation of circle ( π₯ β 3 ) + ( π¦ β 1 ) = 1 0 0 ο¨ ο¨ by ( β 9 , 4 ) . What is its equation?

Q4:

A circle is the translation of circle ( π₯ + 1 ) + ( π¦ + 1 ) = 4 9 ο¨ ο¨ by ( 4 , β 5 ) . What is its equation?

Q5:

A circle is the translation of circle ( π₯ + 3 ) + ( π¦ + 4 ) = 1 0 0 ο¨ ο¨ by ( β 6 , β 1 ) . What is its equation?

Q6:

Given π΄ ( β 2 , β 4 ) and π΅ ( β 1 0 , 0 ) , determine the equation of the circle for which π΄ π΅ is a diameter.

Q7:

The circle in the figure has equation π₯ + π¦ β 1 6 π₯ + 2 π¦ + 4 9 = 0 ο¨ ο¨ , and the line πΏ is π₯ + π¦ + 1 = 0 . Find the length of π΄ π΅ to the nearest hundredth.

Q8:

Given that ( π + 1 0 ) π₯ + π¦ β 1 6 π¦ + ( π + 2 3 ) π₯ π¦ + ( π β π ) π₯ + 1 0 9 = 0 2 2 is a circle, what is its radius?

Q9:

Determine the equation of the circle that passes through ( 1 , β 5 ) and ( 5 , β 5 ) and has its centre on the π₯ -axis.

Q10:

Find the equation of the circle whose centre ( 2 , 7 ) , touching the π₯ -axis.

Q11:

What are the relative positions of circles ( π₯ β 5 ) + ( π¦ + 1 4 ) = 9 2 2 and ( π₯ β 1 3 ) + ( π¦ + 1 4 ) = 3 6 2 2 ?

Q12:

The equation 7 π₯ + π π¦ + π π₯ π¦ β 9 = 0 2 2 represents a circle, what are π and π ?

Q13:

Find the equations of all possible circles that pass through the two points π΄ ( β 7 , 0 ) and π΅ ( 1 , 0 ) and have a radius of 5.

Q14:

Consider a circle centered at ( 1 , 1 ) passing through the origin.

Write the equation of the circle.

The circle is dilated by a scale factor of two such that the circle still passes through the origin. Write the equation of the circle.

What is the radius of the new circle?

Q15:

Does the equation π₯ + π¦ β 4 π₯ β 8 π¦ β 6 1 = 0 2 2 represent a circle?

Q16:

Does the equation π₯ + π¦ β 1 6 π₯ β 1 0 π¦ + 1 8 9 = 0 2 2 represent a circle?

Q17:

In what interval must β lie for π₯ + π¦ + 2 π₯ β 4 π¦ β β + 1 4 = 0 2 2 to be the equation of a circle?

Q18:

Find all the real values of β for which π₯ + π¦ + 4 π₯ + 2 π¦ + β β 7 β β 3 = 0 2 2 2 is the equation of a circle.

Q19:

The line π¦ = π π₯ is tangent to the circle ( π₯ β 5 ) + ( π¦ β 3 ) = 2 5 2 2 . What is π ?

Q20:

Given that the equation of the given circle is ( π₯ β 4 ) + ( π¦ β 9 ) = 2 2 5 ο¨ ο¨ , find the length of π΄ π΅ .

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