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Revision Notes on Parabola

  • The general equation of a conic is ax2 + 2hxy + by2 + 2gx + 2fy +c =0. Here if e =1 and D≠ 0, then it represents a parabola. 

  • The general equation of parabola is (y-y0)2 = (x-x0), which has its vertex at (x0, y0). 

  • The general equation of parabola with vertex at (0, 0) is given by y2 = 4ax, and it opens rightwards.

  • The parabolax2 = 4ay opens upwards.

  • The equation y2 = 4ax is considered to be the standard equation of the parabola for which the various components are

  1. Parabolay2 = 4axVertex at (0,0) 

  2. Directrix is x+a = 0

  3. Axis is y = 0

  4. Focus is (a, 0)

  5. Length of latus rectum = 4a

  6. Ends of latus rectum are L(a, 2a) and L’(a, -2a)

  • The parabola y = a(xh)2 + khas its vertex at (h, k)

  • The perpendicular distance from focus on directrix is half the length of latus rectum

  • Vertex is the middle point of the focus and the point of intersection of directrix and axis

  • Two parabolas are said to be equal if they have the same latus rectum

  • The point (x1, y1) lies outside, on or inside the parabolay2 = 4ax, according as the expression y12 = 4ax1 is positive, zero or negative.

  • Length of the chord intercepted by the parabola on the line y = mx + c is (4/m2) √a(1+m2) (a-mc)

  • Length of the focal chord which makes an angle δ with the x-axis is 4a cosec2δ

  • In parametric form, the parabola is represented by the equations x = at2 and y =2at

  • The equation of a chord joining t1 and t2 is given by 2x – (t1 + t2) y + 2at1t2 = 0

  • If a chord joining t1, t2 and t3, t4 pass through a point (c, 0) on the axis, then t1t2 = t3t4 = -c/a

  • Tangents to the parabola y2 = 4ax

  1. yy1 = 2a(x+x1) at the point (x1, y1)

  2. y = mx + a/m ( m ≠ 0) at (a/m2, 2a/m)

  3. ty = x+at2 at (at2, 2at)

  • Normals to the parabola y2 = 4ax

  1. y-y1 = -y1/ 2a(x-x1) at the point (x1, y1)

  2. y = mx -2am – am3 at (am2, -2am)

  3. y + tx = 2at +at3 at (at2, 2at)

  • The equation of the director circle to the parabola is x + a = 0 which is same as the equation of the directrix

  • The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus.

  • The orthocenter of any triangle formed by three tangents to a parabola y2 = 4ax lies on then directrix and has the coordinates –a, a(t1 + t2 + t3 + t1t2t3).

  • The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.

  • A circle circumscribing the triangle formed by three co-normal points passes through the vertex of the parabola and its equation is given by

2(x2 + y2)– 2(h+2a)x - ky =0

The two vital parabolas along with their basic components like vertex and directrix are tabulated below:

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