The right bisector of a line segment bisects the line segment at \(90°. \)
The endpoints of the line segment are given as \(A (3, 4) \) and \(B (-1, 2).\)
Accordingly, mid-point of
\(AB = \left(\frac{3-1}{2},\frac{4+2}{2}\right)=(1,3)\)
Slope of AB \(\frac{2-4}{-1-3}=\frac{-2}{-4}=\frac{1}{2}\)
∴ Slope of the line perpendicular to \(AB =\frac{-1}{(\frac{1}{2})}=-2\)
The equation of the line passing through \((1, 3)\) and having a slope of -2 is
\((y - 3) = -2 (x - 1) \)
\(y - 3 = -2x + 2\)
\(2x + y = 5\)
Thus, the required equation of the line is \(2x + y = 5.\)
If \( (a, b) \) be the orthocenter of the triangle whose vertices are \( (1, 2) \), \( (2, 3) \), and \( (3, 1) \), and \( I_1 = \int_a^b x \sin(4x - x^2) dx \), \( I_2 = \int_a^b \sin(4x - x^2) dx \), then \( 36 \frac{I_1}{I_2} \) is equal to: