The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x – y = 3, respectively. If its orthocentre is \((2, a),−\frac{1}{2}<a<2 \) then p is equal to _______.
Slope of AH \(=\frac{a+2}{1}\) Slope of BC\(=−\frac{1}{p}\) ∴ p = a + 2 …(i) Coordinate of C \(=(\frac{18p−30}{p+1},\frac{15p−33}{p+1})\) Slope of HC \(=\frac{\frac{15p−33}{p+1}−a}{\frac{18p−30}{p+1}−2}\) \(=\frac{15p−33−(p−2)(p+1)}{18p−30−2p−2}\) \(=\frac{16p−p^2−31}{16p−32}\) \(∵ \frac{16p−p^2−31}{16p−32}×−2=−1\) ∴ p2 – 8p + 15 = 0 ∴ p = 3 or 5 But if p = 5 then a = 3 not acceptable ∴ p = 3 So, the answer is 3.
Coordinate geometry, also known as analytical geometry or Cartesian geometry, is a branch of mathematics that combines algebraic techniques with the principles of geometry. It provides a way to represent geometric figures and solve problems using algebraic equations and coordinate systems. The central idea in coordinate geometry is to assign numerical coordinates to points in a plane or space, which allows us to describe their positions and relationships using algebraic equations. The most common coordinate system is the Cartesian coordinate system, named after the French mathematician and philosopher René Descartes.
