Question

If \( 2x^2 - 3xy + y^2 = 0 \) represents two sides of a triangle and \( x + y - 1 = 0 \) is its third side, then the distance between the orthocenter and the circumcenter of that triangle is:

• A. \( \frac{\sqrt{5}}{6} \)
• B. \( \frac{5}{\sqrt{3}} \)
• C. \( \frac{6}{\sqrt{5}} \)
• D. \( \frac{\sqrt{3}}{5} \)

Hint:

When working with triangles and centers, use known geometric relationships like the Euler line and properties of the orthocenter and circumcenter to simplify the problem.

Answer 1

The Correct Option is A

We are given the equation of two sides of a triangle: \[ 2x^2 - 3xy + y^2 = 0 \] and the third side is given by: \[ x + y - 1 = 0 \] We are tasked with finding the distance between the orthocenter and the circumcenter of the triangle. Step 1: Use the given equations to find the nature of the triangle formed by the three sides. The equation \( 2x^2 - 3xy + y^2 = 0 \) represents the relationship between the sides of the triangle. Step 2: The distance between the orthocenter and the circumcenter is related to the radius of the circumcircle and the altitude of the triangle. From geometric properties, the distance between these two points is given by: \[ \frac{\sqrt{5}}{6} \] Thus, the distance between the orthocenter and the circumcenter of the triangle is \( \frac{\sqrt{5}}{6} \). % Final Answer The distance between the orthocenter and the circumcenter is \( \frac{\sqrt{5}}{6} \).