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How to find the point of a circle closest to the origin?

Question:

How to find the point of a circle closest to the origin?

Circle

The collection of all the points which are at equal distance from a fixed point is known as Circle. the fixed point is called the center of the circle and the fixed distance is called the radius of the circle. The standard equation of the circle is {eq}(x-h)^2+(y-k)^2=r^2 {/eq} Where (h,k) are the coordinates of the center of the circle and r is the radius of the circle.

Answer and Explanation:

The standard equation of the circle is {eq}(x-h)^2+(y-k)^2=r^2 {/eq} Where (h,k) are the coordinates of the center of the circle and r is the radius of the circle.

To find the point on the circle which is closest to the origin, we can follow the following steps:

Step1 Construct a distance function which is equal to the distance between the general point (x,y) on the circle and the point origin (0,0)

i.e. {eq}D=\sqrt{(x-0)^2+(y-0)^2} \\ =\sqrt{x^2+y^2} {/eq}

Step2 Now make the distance function D of one variable i.e. use the equation of the circle to substitute the value of any variable in the equation of the distance function.

Step3 Differentiate the distance function with respect to that independent variable and find the critical points.

Step4Use the first derivative test to find the extrema of the distance function.

Step5 The point where the distance function will minimum will be the closest point to the origin.


Learn more about this topic:

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