If the line y = x touches the parabola y = x² + bx + c at the point (1, 1), then

(A)   b = 0, c = −1

(B)   b = −1, c = Z

(C)   b = −1, c = −1

(D)  b = −1, c = 1

Determine the coordinates of the focus, the equation of the directrix and the length of the latus rectum of the parabola

y² = (8/3)x.

Passage: For the parabola y² = 4ax, the vertex is (0, 0), the focus is (a, 0) and the directrix is x + a = 0.

 Answer the following three questions.

(i) The vertex of the parabola (y – 1)² = 2(x – 1) is 

(A)  (1,0)

(B)  (2,0) 

(C)  (1,1)

(D)  (0,1)

(ii)  Focus of the parabola y² = 4(x – 1) is

(A)  (1, 0)

(B)  (2, 0)

(C)  (1, 1)

(D)  (2, 2)

(iii)  The directrix of the parabola (y – 2)² = 4(x – 1)

(A)   x = 1

(B)   x = -1

(C)   x = 2 

(D)  x =  0

Show that the locus of the midpoints of chords of a parabola passing through the vertex is in turn a parabola whose latus rectum is half of the latus rectum of the original. 

Equation of the normal at t is

tx + y = 2at + at³

In general, three normals can be drawn through a point in the plane of a parabola to the curve.

Find the director circle of the parabola x² + 2y = 4x – 3 .