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Problems

  1. Assume that \(m\) is not negative. The vertex Q of parabola \(y = x^2 - 2(m+1)x - (m+3)\) is on the line \(y = -2x - 2\). The parabolic curve crosses x-axis at two different points A and B. A is on the left of B. (1) What are the coordinates of A, B, and Q? (2) Let \(P=(1,1)\). Prove PA is perpenticular to the line \(y = -2x - 2\).

  2. Assume that the parabola \(y = x^2 - 2x -8\) crosses x-axis at A and B, its vertex is P. What is area of the triangle \(\bigtriangleup ABP\)?

  3. In Figure 1, the parabolic curve \(y=ax^2 + bx +c\) crosses x-axis at A and B, and crosses y-axis at C. \(OB=OC=4OA\), the area of \(\bigtriangleup ABC = 40\). (1) What are the coordinates of A, B and C? (2) What are a, b, and c?


Figure 1: Figure 1

  1. Assume that parabolic curve \(y = -x^2 + 2(m+1)x + m + 3\) crosses x-axis at two different points A and B. A is on the positive sidie. B is on the negative side. What is the range of \(m\)?

Solutions

  1. A(-1, 0), B(3, 0), Q(1, -4)

  1. 27

  2. A(-2, 0), B(8, 0), C(0, 8), a=-1/2, b=3, c=8

  3. m > -3

Set4