what is the distance from the center of a circle to a point on the circle

Shortest Distance between a Indicate and a Circle

What is the distance between a circle $C$ with equation $x^2+y^2=r^2$ which is centered at the origin and a point $P\left({\mathrm{10}}_1,y_{\mathrm{ane}}\right)$ ?

The ray $\overrightarrow{OP}$ , starting at the origin $O$ and passing through the point $P$ , intersects the circle at the point closest to $P$ . So, the altitude between the circle and the bespeak will be the difference of the altitude of the point from the origin and the radius of the circle.

Using the Distance Formula , the shortest distance between the point and the circumvolve is $\left|\sqrt{{\left(x_1\right)}^{\mathrm{ii}}+{\left(y_1\right)}^{\mathrm{ii}}}-r\right|$ .

Note that the formula works whether $P$ is within or outside the circle.

If the circle is not centered at the origin but has a heart say $\left(h,k\right)$ and a radius $r$ , the shortest distance between the bespeak $P\left(x_1,y_1\right)$ and the circumvolve is $\left|\sqrt{{\left(x_1-h\right)}^2+{\left(y_1-k\right)}^{\mathrm{ii}}}-r\right|$ .

Instance 1:

What is the shortest distance between the circle ${\mathrm{ten}}^{\mathrm{ii}}+y^2=9$ and the point $A(3,4)$ ?

The circumvolve is centered at the origin and has a radius $\mathrm{three}$ .

So, the shortest distance $D$ between the point and the circle is given past

$\begin{array}{l}D=\left|\sqrt{{\left(\mathrm{three}\right)}^{\mathrm{two}}+{\left(4\right)}^{\mathrm{two}}}-\mathrm{three}\right|\\ =\left|\sqrt{25}-3\right|\\ =\left|\mathrm{v}-3\right|\\ =\mathrm{ii}\end{array}$

That is, the shortest distance between them is $\mathrm{ii}$ units.

Instance 2:

What is the shortest altitude betwixt the circle $x^2+y^2=36$ and the point $Q(-2,\mathrm{two})$ ?

The circle is centered at the origin and has a radius $\mathrm{vi}$ .

And then, the shortest altitude $D$ between the point and the circle is given by

$\begin{array}{l}D=\left|\sqrt{{(-2)}^2+{\left(2\right)}^2}-\mathrm{vi}\right|\\ =\left|\sqrt{\mathrm{eight}}-6\right|\\ =6-2\sqrt{2}\\ \approx \mathrm{iii.17}\end{array}$

That is, the shortest distance between them is well-nigh $3.17$ units.

Example three:

What is the shortest distance between the circumvolve ${\left(x+\mathrm{iii}\right)}^2+{\left(y-3\right)}^{\mathrm{two}}=5^2$ and the point $Z(-2,0)$ ?

Compare the given equation with the standard grade of equation of the circumvolve,

${\left(x-h\right)}^{\mathrm{ii}}+{\left(y-k\right)}^2=r^{\mathrm{ii}}$ where $(h,\mathrm{1000})$ is the center and $r$ is the radius.

The given circle has its eye at

$(-\mathrm{three},3)$

and has a radius of

$\mathrm{v}$

units.

And then, the shortest altitude $D$ between the point and the circumvolve is given by

$\begin{array}{l}D=\left|\mathrm{v}-\sqrt{{\left(-3-\left(-2\right)\right)}^2+{\left(3-0\right)}^2}\right|\\ =\left|\mathrm{v}-\sqrt{1+\mathrm{ix}}\right|\\ =\left|\mathrm{five}-\sqrt{10}\right|\\ \approx 1.84\end{array}$

That is, the shortest distance between them is almost $\mathrm{i.84}$ units.

Example 4:

What is the shortest altitude between the circumvolve ${\mathrm{10}}^{\mathrm{ii}}+y^2-8x+10y-8=0$ and the point $P\left(-4,-\mathrm{xi}\right)$ ?

Rewrite the equation of the circle in the grade ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ where $(h,k)$ is the center and $r$ is the radius.

$\begin{array}{l}{x}^{\mathrm{ii}}+y^2-8x+10y-8=0\\ {\mathrm{ten}}^2-8\mathrm{ten}+\mathrm{xvi}+y^{\mathrm{two}}+10y+25=8+16+25\\ {\left(\mathrm{10}-4\right)}^2+{\left(y+5\right)}^{\mathrm{two}}=49\\ {\left(x-4\right)}^2+{\left(y+5\right)}^2=7^{\mathrm{two}}\end{array}$

So, the circle has its eye at

$(4,-5)$

and has a radius of

$7$

units.

Then, the shortest distance $D$ between the point and the circle is given by

$\begin{array}{c}D=\left|\sqrt{{\left(-\mathrm{four}-4\right)}^2+{\left(-11-\left(-5\right)\right)}^{\mathrm{ii}}}-7\right|\\ =\left|\sqrt{64+36}-7\right|\\ =\sqrt{100}-7\\ =3\end{array}$

That is, the shortest altitude between them is $3$ units.

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Source: https://www.varsitytutors.com/hotmath/hotmath_help/topics/shortest-distance-between-a-point-and-a-circle

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