Conic Sections · Question 10

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?

1. The length of the line segment OA1 is 4 3
2. The length of the line segment A1B1 is 16
3. The orthocenter of the triangle A1B1C1 is (0,0)
4. The orthocenter of the triangle A1B1C1 is (1,0)

Answer

(1) The length of the line segment OA1 is 4 3, (3) The orthocenter of the triangle A1B1C1 is (0,0)

Solution

For y2=8x (a=2), tangent y=mx+ 2m touches at ( 2m2, 4m ).

Through (-4,0): 0=-4m+ 2m m2= 12, so A1,B1=(4, 4 2).

OA1= 16+32=4 3; A1B1=8 2 16.

By symmetry the orthocenter is on the x-axis. Slope of B1C1= -4 28=- 1 2, so altitude from A1: y-4 2= 2(x-4); at y=0, x=0.

Orthocenter =(0,0). Answer: (A), (C)