First slide

Equation of line in Straight lines

Question

$The points (2, 5), (2, 7), (1, 7) forms a triangle. The inclination of the line joining the$ $ortho centre and the circum centre of the triangle is$

Moderate

A

$\tan^{- 1} (3)$
B

$\tan^{- 1} (1)$
C

$\tan^{- 1} (2)$
D

$\tan^{- 1} (4)$

Solution

$Since the x -coordinates of two points A (2, 5), B (2, 7) are same, the line joining A, B is vertical line$
$And since y -coordinates of two points B (2, 7), C (1, 7) are same. the line joining B, C is$
$horizontal line, so that the triangle A B C is right angled triangle with vertex at right angle is$ $at B (2, 7)$
In right angled triangle, the ortho centre is vertex at right angle and the circum centre is midpoint of the hypotenuse
$Hence, the circum centre is S = (\frac{2 + 1}{2}, \frac{5 + 7}{2}) = (\frac{3}{2}, 6)$
$The ortho centre is O (2, 7)$
The slope of the line joining ortho centre and the circum centre is m
$\begin{matrix} m = \frac{6 - 7}{\frac{3}{2} - 2} \\ = \frac{- 1}{- \frac{1}{2}} \\ = 2 \end{matrix}$
$If the inclination of any line θ then the slope of that line is m = \tan θ$
$Here, \tan θ = 2 \Rightarrow θ = \tan^{- 1} (2)$
$Therefore, the inclination of the line joining circum centre and ortho centre is \tan^{- 1} (2)$

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