# The centroid of the triangle is $\left( {3,3} \right)$and the orthocenter is$\left( {3, - 5} \right)$, then its circumcenter is

$

{\text{a}}{\text{. }}\left( {0,4} \right) \\

{\text{b}}{\text{. }}\left( {0,8} \right) \\

{\text{c}}{\text{. }}\left( {6,2} \right) \\

{\text{d}}{\text{. }}\left( {6, - 2} \right) \\

$

Last updated date: 21st Mar 2023

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Answer

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Hint: - Centroid divides the line joining orthocenter and circumcenter in the ratio$2:1$

As we know the centroid divides the line joining orthocenter and circumcenter in the ratio$2:1$

Given orthocenter coordinates is $\left( {3, - 5} \right)$and centroid coordinates is$\left( {3,3} \right)$

Let the circumcenter be$\left( {x,y} \right)$

Then by section formula the coordinates of centroid is written as

$3 = \frac{{mx - 3n}}{{m + n}},{\text{ }}3 = \frac{{my - \left( { - 5} \right)n}}{{m + n}}$

Here$m = 2$and$n = 1$

\[

3 = \frac{{2x - 3}}{{2 + 1}},{\text{ }}3 = \frac{{2y + 5}}{{2 + 1}} \\

\Rightarrow 9 = 2x - 3,{\text{ }}9 = 2y + 5 \\

\Rightarrow x = \frac{{12}}{2} = 6 \\

\Rightarrow y = \frac{4}{2} = 2 \\

\]

Therefore the coordinates of circumcenter is$\left( {6,2} \right)$

Hence option c is correct.

Note: - In such types of question the key concept we have to remember is that the centroid divides the line joining orthocenter and circumcenter in the ratio$2:1$, then apply the section formula which is stated above, so after simplification we will get the required coordinates of the circumcenter.

As we know the centroid divides the line joining orthocenter and circumcenter in the ratio$2:1$

Given orthocenter coordinates is $\left( {3, - 5} \right)$and centroid coordinates is$\left( {3,3} \right)$

Let the circumcenter be$\left( {x,y} \right)$

Then by section formula the coordinates of centroid is written as

$3 = \frac{{mx - 3n}}{{m + n}},{\text{ }}3 = \frac{{my - \left( { - 5} \right)n}}{{m + n}}$

Here$m = 2$and$n = 1$

\[

3 = \frac{{2x - 3}}{{2 + 1}},{\text{ }}3 = \frac{{2y + 5}}{{2 + 1}} \\

\Rightarrow 9 = 2x - 3,{\text{ }}9 = 2y + 5 \\

\Rightarrow x = \frac{{12}}{2} = 6 \\

\Rightarrow y = \frac{4}{2} = 2 \\

\]

Therefore the coordinates of circumcenter is$\left( {6,2} \right)$

Hence option c is correct.

Note: - In such types of question the key concept we have to remember is that the centroid divides the line joining orthocenter and circumcenter in the ratio$2:1$, then apply the section formula which is stated above, so after simplification we will get the required coordinates of the circumcenter.

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