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.

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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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You are watching: Find the shortest distance, d, from the point (3, 0, −2) to the plane x + y + z = 2.

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 .

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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 .

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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.

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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.

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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