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Question:
Consider a triangle \(Δ\) whose two sides lie on the x-axis and the line \(x + y + 1 = 0\). If the orthocenter of \(Δ\) is (1, 1), then the equation of the circle passing through the vertices of the triangle \(Δ\) is;
Consider a triangle \(Δ\) whose two sides lie on the x-axis and the line \(x + y + 1 = 0\). If the orthocenter of \(Δ\) is (1, 1), then the equation of the circle passing through the vertices of the triangle \(Δ\) is;
Updated On: Aug 6, 2024
x2 + y2 – 3x + y = 0
x2 + y2 + x + 3y = 0
x2 + y2 + 2y – 1 = 0
x2 + y2 + x + y = 0
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The Correct Option is B
Solution and Explanation
Given that the mirror image of the orthocenter lies on the circumcircle:
The image of the point (1, 1) reflected over the x-axis is (1, -1). The image of the point (1, 1) reflected over the line \(x+y+1\)=0 is (-2, -2).
Therefore, the circle passing through both (1, -1) and (-2, -2) is determined.
Thus, the circle represented by the equation x2 + y2 + x + 3y = 0 satisfies this condition.
Hence the correct option is B
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Concepts Used:
Coordinate Geometry
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.
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.
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