Question
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?
JEE Advanced
Mathematics
Parabola
2024
JEE Advanced 2024 (Paper 2)
JEE Advanced Mathematics Question (2024) — Solution
Options
- A. The length of the line segment O A_1 is 4 3
- B. The length of the line segment A_1 B_1 is 16
- C. The orthocenter of the triangle A_1 B_1 C_1 is (0,0)
- D. The orthocenter of the triangle A_1 B_1 C_1 is (1,0)
Answer
C. The orthocenter of the triangle A_1 B_1 C_1 is (0,0)
Step-by-step solution
Equation of tangent at (2 t^2, 4 t ) is ty = x +2 t ^2 It is passing through (-4,0) 0=-4+2 t ^2 t = 2 . array rl A _1=(4,4 2 ) \\ B _1=(4,-4 2 ) array \ aligned & OA _1= 48 =4 3 \\ & ~A _1 ~B _1=8 2 aligned Equation of altitude of A_1 B_1 C_1 drawn from A_1 is y-4 2 = 2 (x-4) 2 x-y=0 ...(1) Equation of altitude of A_1 B_1 C_1 drawn from C_1 is x =0 ...(2) Solving (1) and (2) orthocentre is (0,0) correct options are (1), (3)
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