Question

Let \( A_1, B_1, C_1 \) be three points in the \( xy \)-plane. Suppose that the lines \( A_1C_1 \) and \( B_1C_1 \) are tangents to the curve \( y^2 = 8x \) at \( A_1 \) and \( B_1 \), respectively. If \( O = (0, 0) \) and \( C_1 = (-4, 0) \), then which of the following statements is (are) TRUE?

• A. The length of the line segment \( OA_1 \) is \( 4\sqrt{3} \).
• B. The length of the line segment \( A_1B_1 \) is \( 16 \).
• C. The orthocenter of the triangle \( A_1B_1C_1 \) is \( (0, 0) \).
• D. The orthocenter of the triangle \( A_1B_1C_1 \) is \( (1, 0) \).

Hint:

For tangents to parabolas, parameterize the points and apply geometric properties of triangles.

Answer 1

The Correct Option is A

Let \( A_1 = (2t_1^2, 4t_1) \) and \( B_1 = (2t_2^2, 4t_2) \). Given \( C_1 = (-4, 0) \), we know: \[ t_1 = -\sqrt{2}, \quad t_2 = \sqrt{2}. \] Coordinates: \[ A_1 = (4, -4\sqrt{2}), \quad B_1 = (4, 4\sqrt{2}). \] Length of \( OA_1 \): \[ OA_1 = \sqrt{(4-0)^2 + (-4\sqrt{2}-0)^2} = 4\sqrt{3}. \] Length of \( A_1B_1 \): \[ A_1B_1 = \sqrt{(4-4)^2 + (4\sqrt{2} + 4\sqrt{2})^2} = 16. \] The orthocenter of \( \triangle A_1B_1C_1 \) is \( (0, 0) \), verified using altitude equations.