Conic Sections

Let T be the tangent to the parabola y2=16x at the point (64,32). Let L be the tangent to the same parabola at another point (x1, y1) on the parabola. If L and T are perpendicular to each other, then the distance between the point (x1, y1) and the focus of the parabola, is

1. 154
2. 4
3. 174
4. 5

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)

A normal with slope 1 6 is drawn from the point (0,- ) to the parabola x2=-4ay, where a>0. Let L be the line passing through (0,- ) and parallel to the directrix of the parabola. Suppose that L intersects the parabola at two points A and B. Let r denote the length of the latus rectum and s denote the square of the length of the line segment AB. If r:s=1:16, then the value of 24a is .