Let \(A_1, B_1, C_1\) be three points in the \(x y\)-plane. Suppose that the lines \(A_1 C_1\) and \(B_1 C_1\) are tangents to the curve \(y^2=8 x\) 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?

Equation of tangent at \(\left(2 t^2, 4 t\right)\) is
\(\mathrm{ty}=\mathrm{x}+2 \mathrm{t}^2\)
\(\because\) It is passing through \((-4,0)\)
\(\therefore 0=-4+2 \mathrm{t}^2 \Rightarrow \mathrm{t}= \pm \sqrt{2}\)
\(\left.\begin{array}{rl}\therefore \mathrm{A}_1=(4,4 \sqrt{2}) \\ \mathrm{B}_1=(4,-4 \sqrt{2})\end{array}\right\} \begin{aligned} & \mathrm{OA}_1=\sqrt{48}=4 \sqrt{3} \\ & \mathrm{~A}_1 \mathrm{~B}_1=8 \sqrt{2}\end{aligned}\)
Equation of altitude of \(\Delta A_1 B_1 C_1\) drawn from \(A_1\) is
\(y-4 \sqrt{2}=\sqrt{2}(x-4)\)
\(\Rightarrow \sqrt{2} x-y=0\) ...(1)
Equation of altitude of \(\Delta A_1 B_1 C_1\) drawn from \(C_1\) is
\(\mathrm{x}=0\) ...(2)
Solving (1) and (2) \(\Rightarrow\) orthocentre is \((0,0)\)
\(\therefore\) correct options are (1), (3)