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.