Coordinate Geometry Practice: Triangle, Straight Line, Ellipse, Equilateral Triangle

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Questions and Answers

What is the relationship between the Arithmetic Mean (A.M), Geometric Mean (G.M), and Harmonic Mean (H.M) as stated in the given text?

The text states that A.M ≥ G.M ≥ H.M.

What are the coordinates of the third vertex of a triangle if two of its vertices are (0, -4) and (6, 0), and the medians meet at the point (2, 0)?

The third vertex of the triangle is (2, -4).

What is the equation of the straight line that bisects the line segment joining the points (3, -4) and (1, 2) at a right angle?

The equation of the straight line is $y = (2/5)x + (8/5)$.

What is the equation of the ellipse whose center is at (0, 0), one vertex is at (0, -5), and one end of the minor axis is at (3, 0)?

<p>The equation of the ellipse is $rac{x^2}{9} + rac{y^2}{25} = 1$.</p> Signup and view all the answers

Prove that the points (a, b, c), (b, c, a), and (c, b, a) form an equilateral triangle.

<p>To prove that the points (a, b, c), (b, c, a), and (c, b, a) form an equilateral triangle, we need to show that the distances between any two of these points are equal.</p> Signup and view all the answers

Explain the relationship between the Arithmetic Mean (A.M), Geometric Mean (G.M), and Harmonic Mean (H.M) as stated in the given text.

<p>The given text states that the relationship between the Arithmetic Mean (A.M), Geometric Mean (G.M), and Harmonic Mean (H.M) is: A.M ≥ G.M ≥ H.M.</p> Signup and view all the answers

Derive the equation of the straight line that bisects the line segment joining the points (3, -4) and (1, 2) at a right angle.

<p>The equation of the straight line that bisects the line segment joining the points (3, -4) and (1, 2) at a right angle is $y = -rac{1}{3}x + rac{5}{3}$.</p> Signup and view all the answers

Write the equation of the ellipse whose center is at (0, 0), one vertex is at (0, -5), and one end of the minor axis is at (3, 0).

<p>The equation of the ellipse is $rac{x^2}{9} + rac{y^2}{25} = 1$.</p> Signup and view all the answers

Prove that the points (a, b, c), (b, c, a), and (c, b, a) form an equilateral triangle.

<p>To prove that the points (a, b, c), (b, c, a), and (c, b, a) form an equilateral triangle, we need to show that the distances between any two of these points are equal.</p> Signup and view all the answers

Find the coordinates of the third vertex of a triangle if two of its vertices are (0, -4) and (6, 0), and the medians meet at the point (2, 0).

<p>The coordinates of the third vertex of the triangle are (4, 2).</p> Signup and view all the answers

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