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Circumcentre and Orthocentre - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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  • Circumcentre and Orthocentre is considered one of the most asked concept.

  • 51 Questions around this concept.

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The radical centre of three circles described on the three sides of a triangle as diameter is the 

 

 

Let the position vectors of the vertices $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ of a triangle be $2 \hat{\imath}+2 \hat{\jmath}+\hat{k}, \hat{\imath}+2 \hat{\jmath}+2 \hat{k}$ and $2 \hat{\imath}+\hat{\jmath}+2 \hat{k}$ respectively. Let $\ell_1, \ell_2$ and $\ell_3$ be the lengths of perpendiculars drawn from the ortho center of the triangle on the sides $\mathrm{AB}, \mathrm{BC}$ and $\mathrm{CA}$ respectively, then $\ell_1^2+\ell_2^2+\ell_3^2$ equals:

Concepts Covered - 1

Circumcentre and Orthocentre

Circumcentre

Perpendicular bisector of a side of a triabgle is the line through the midpoint of a side and perpendicular to it.  

The Circumcentre (O) of a triangle is the point of intersection of the perpendicular bisectors of the sides of a triangle.

Circumcentre is also defined as the center of a circle that passes through the vertices of a given triangle.

Coordinates of Circumcentre (O) is

\\\mathrm{\mathbf{\left (\frac{x_1\sin2A+x_2\sin2B+x_3\sin2C}{\sin2A+\sin2B+\sin2C},\;\;\frac{y_1\sin2A+y_2\sin2B+y_3\sin2C}{\sin2A+\sin2B+\sin2C}\right )}}

Note:

For a right angled triangle, the circumcenter is the mid point of the hypotenuse.

 

Orthocentre:

The Orthocentre (H) of a triangle is the point of intersection of altitudes which are drawn from one vertex to the opposite side of a triangle. 

Coordinates of Orthocentre (H) is

\\\mathrm{\mathbf{\left (\frac{x_1\tan A+x_2\tan B+x_3\tan C}{\tan A+\tan B+\tan C},\;\;\frac{y_1\tan A+y_2\tan B+y_3\tan C}{\tan A+\tan B+\tan C}\right )}}

 

Note:

For a right angled triangle, the orthocenter is the vertex containing the right angle 

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Circumcentre and Orthocentre

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Circumcentre and Orthocentre

Mathematics for Joint Entrance Examination JEE (Advanced) : Coordinate Geometry

Page No. : 1.11

Line : 31

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