The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 39 and x – y = 3, respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true?
The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 39 and x – y = 3, respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true?
\((AC)^2 = 9p\)
\((AC)^2 + p^2 = 136\)
\(32 < area (ΔABC)<36\)
\(34<area(ΔABC)<38\)
The Correct Option is D
Solution and Explanation
The correct answer is (D) : \(34<area(ΔABC)<38\)
Intersection of 2x + y = 0 and x – y = 3 :A(1, –2)
Equation of perpendicular bisector of AB is
x – 2y = –4
Equation of perpendicular bisector of AC is
x + y = 5
Point B is the image of A in line x – 2y + 4 = 0
which can be obtained as
\(B(\frac{-13}{5},\frac{26}{5})\)
Similarly vertex C : (7, 4)
Equation of line BC : x + 8y = 39
So, p = 8
\(AC = \sqrt{(7-1)^2+(4+2)^2}\)
\(= 6\sqrt2\)
Area of triangle ABC = 32.4
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Concepts Used:
Coordinate Geometry
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.