Parabola

If the line lx + my + n = 0 touches the parabola y² = 4ax, then

(a)   In = am

(b)  In² = am²

(c)  l²n = am²

(d)   In = am²

The orthocentre of a triangle formed by three tangents to a parabola y² = 8x lies on

(a)  x + 1 = 0

(b)  x + 2 = 0

(c)  x + = 0

(d)  x – 1 = 0

Show that the chords of contacts of points on the line 2x – 3y + 4 = 0 with respect to the parabola y² = 4ax pass through a fixed point.

If P is a point on the line x + 4a = 0 and QR is the chord of contact of P with respect to y² = 4ax, then ∠QOR (where O is the vertex) is equal to

 (A)  45°

(B)  60°

(C)  30°

(D)  90°

Equation of the normal at t is

tx + y = 2at + at³

If two tangents to a parabola make complementary angles with the axis, then show that their point of intersection lies on the line x = a.

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

Find the vertex, focus, directrix and latus rectum of the parabola y² + 4x – 2y + 3 = 0.