What is the distance in between a circle C through equation x 2 + y 2 = r 2 which is focused at the origin and a point P ( x 1 , y 1 ) ?

The ray O P → , beginning at the beginning O and also passing with the suggest P , intersects the circle at the suggest closest to P . So, the distance between the circle and also the point will certainly be the distinction of the distance of the allude from the beginning and also the radius of the circle.

Using the Distance Formula , the shortest distance in between the allude and the circle is | ( x 1 ) 2 + ( y 1 ) 2 − r | .

Note that the formula functions whether P is inside or exterior the circle.

If the circle is not centered at the beginning but has a center say ( h , k ) and a radius r , the shortest distance in between the allude P ( x 1 , y 1 ) and the circle is | ( x 1 − h ) 2 + ( y 1 − k ) 2 − r | .

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** Example 1: **

What is the shortest distance between the circle x 2 + y 2 = 9 and the point A ( 3 , 4 ) ?

The circle is centered at the beginning and also has actually a radius 3 .

So, the shortest distance D in between the allude and also the circle is given by

D = | ( 3 ) 2 + ( 4 ) 2 − 3 | = | 25 − 3 | = | 5 − 3 | = 2

That is, the shortest distance in between them is 2 devices.

** Example 2: **

What is the shortest distance between the circle x 2 + y 2 = 36 and also the allude Q ( − 2 , 2 ) ?

The circle is focused at the beginning and also has a radius 6 .

So, the shortest distance D between the suggest and also the circle is offered by

D = | ( − 2 ) 2 + ( 2 ) 2 − 6 | = | 8 − 6 | = 6 − 2 2 ≈ 3.17

That is, the shortest distance in between them is around 3.17 units.

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** Example 3: **

What is the shortest distance between the circle ( x + 3 ) 2 + ( y − 3 ) 2 = 5 2 and the point Z ( − 2 , 0 ) ?

Compare the offered equation with the conventional form of equation of the circle,

( x − h ) 2 + ( y − k ) 2 = r 2 where ( h , k ) is the facility and also r is the radius.

The given circle has actually its facility at ( − 3 , 3 ) and also has actually a radius of 5 devices.

Then, the shortest distance D in between the suggest and the circle is provided by

D = | 5 − ( − 3 − ( − 2 ) ) 2 + ( 3 − 0 ) 2 | = | 5 − 1 + 9 | = | 5 − 10 | ≈ 1.84

That is, the shortest distance between them is around 1.84 systems.

** Example 4: **

What is the shortest distance between the circle x 2 + y 2 − 8 x + 10 y − 8 = 0 and also the point P ( − 4 , − 11 ) ?

Recreate the equation of the circle in the develop ( x − h ) 2 + ( y − k ) 2 = r 2 where ( h , k ) is the center and r is the radius.

x 2 + y 2 − 8 x + 10 y − 8 = 0 x 2 − 8 x + 16 + y 2 + 10 y + 25 = 8 + 16 + 25 ( x − 4 ) 2 + ( y + 5 ) 2 = 49 ( x − 4 ) 2 + ( y + 5 ) 2 = 7 2

So, the circle has actually its facility at ( 4 , − 5 ) and has actually a radius of 7 devices.

Then, the shortest distance D in between the suggest and the circle is provided by

D = | ( − 4 − 4 ) 2 + ( − 11 − ( − 5 ) ) 2 − 7 | = | 64 + 36 − 7 | = 100 − 7 = 3