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# The centroid of the triangle is $\left( {3,3} \right)$and the orthocenter is$\left( {3, - 5} \right)$, then its circumcenter is${\text{a}}{\text{. }}\left( {0,4} \right) \\ {\text{b}}{\text{. }}\left( {0,8} \right) \\ {\text{c}}{\text{. }}\left( {6,2} \right) \\ {\text{d}}{\text{. }}\left( {6, - 2} \right) \\$

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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.
Last updated date: 17th Sep 2023
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