Let A1, B1, C1 be three points in the xy-plane. Suppose that the lines A1C1 and B1C1 are tangents to the curve y2 = 8x at A1 and B1, respectively. If O = (0, 0) and C1 = (– 4, 0), then which of the following statements is (are) TRUE?

Engineering

Mathematics

Introduction to Determinants

Question

Let A₁, B₁, C₁ be three points in the xy-plane. Suppose that the lines A₁C₁ and B₁C₁ are tangents to the curve y² = 8x at A₁ and B₁, respectively. If O = (0, 0) and C₁ = (– 4, 0), then which of the following statements is (are) TRUE?

The length of the line segment OA₁ is $4 \sqrt{3}$

The length of the line segment A₁B₁ is 16

The orthocenter of the triangle A₁B₁C₁ is (0, 0)

The orthocenter of the triangle A₁B₁C₁ is (1, 0)

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Equation of tangent at (2t², 4t) is

ty = x + 2t²
∵ It is passing through (– 4, 0)

$∴ 0 = - 4 + 2 t^{2} \Rightarrow t = \pm \sqrt{2}$

$\begin{aligned} ∴ A_{1} = (4, 4 \sqrt{2}) \\ B_{1} = (4, - 4 \sqrt{2}) \end{aligned}\} \begin{aligned} {OA}_{1} = \sqrt{48} = 4 \sqrt{3} \\ A_{1} B_{1} = 8 \sqrt{2} \end{aligned}$

Equation of altitude of ΔA₁B₁C₁ drawn from A₁ is

$y - 4 \sqrt{2} = \sqrt{2} (x - 4)$

$\Rightarrow \sqrt{2} x - y = 0$ .....(1)

Equation of altitude of ΔA₁B₁C₁ drawn from C₁ is
x = 0 …(2)
Solving (1) and (2) ⇒ orthocentre is (0, 0)
∴ correct options are (A), (C)

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