Let $A_{1}, B_{1}, C_{1}$ be three points in the $xy$-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?

(A) The length of the line segment $OA_{1}$ is $4\sqrt{3}$; (C) The orthocenter of the triangle $A_{1} B_{1} C_{1}$ is $(0,0)$

(B) The length of the line segment $A_{1}B_{1}$ is 16

(D) The orthocenter of the triangle $A_{1} B_{1} C_{1}$ is $(1,0)$

JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Parabola

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

A_{1}

and

B_{1}

, respectively. If

O = (0, 0)

and

C_{1} = (- 4, 0)

, then which of the following statements is (are) TRUE?

Answer: (A), (C)

For the parabola

y^{2} = 8 x

, a point on it is

(2 t^{2}, 4 t)

and the tangent at this point is

t y = x + 2 t^{2}

.

Since the tangent passes through

C_{1} = (- 4, 0)

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

A_{1} = (4, 42)

and

B_{1} = (4, - 42)

.

(A)

O A_{1} = 16 + 32 = 48 = 43

. ✓

(B)

A_{1} B_{1} = ∣4 2 - (- 42) ∣ = 82 \neq = 16

. ✗

(C,D) The altitude from

A_{1}

perpendicular to

B_{1} C_{1}

has slope

2

(negative reciprocal of slope of

B_{1} C_{1}

y - 42 = 2 (x - 4) \Rightarrow 2 x - y = 0

The altitude from

C_{1}

perpendicular to

A_{1} B_{1}

(which is vertical) is horizontal:

y = 0

.

Intersection:

x = 0, y = 0

, so orthocenter is

(0, 0)

. ✓(C)

Correct answers: (A), (C)

JEE Advanced · 2024 · 26 May 2024 · Paper 2 · One or more correct options