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- The general equation of a conic is ax
^{2}+ 2hxy + by^{2}+ 2gx + 2fy +c =0. Here if e =1 and D≠ 0, then it represents a parabola. - The general equation of parabola is (y-y
_{0})^{2 }= (x-x_{0}), which has its vertex at (x_{0}, y_{0}). - The general equation of parabola with vertex at (0, 0) is given by y
^{2}= 4ax, and it opens rightwards. - The parabola x
^{2}= 4ay opens upwards. - The equation y
^{2}= 4ax is considered to be the standard equation of the parabola for which the various components are1. Vertex 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(x – h)
^{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 (x
_{1}, y_{1}) lies outside, on or inside the parabola y^{2}= 4ax, according as the expression y_{1}^{2}= 4ax_{1}is positive, zero or negative. - Length of the chord intercepted by the parabola on the line y = mx + c is (4/m
^{2}) √a(1+m^{2}) (a-mc) - Length of the focal chord which makes an angle δ with the x-axis is 4a cosec
^{2}δ - In parametric form, the parabola is represented by the equations x = at
_{2}and y =2at - The equation of a chord joining t
_{1}and t_{2}is given by 2x – (t_{1}+ t_{2}) y + 2at_{1}t_{2}= 0 - If a chord joining t
_{1}, t_{2}and t_{3}, t_{4}pass through a point (c, 0) on the axis, then t_{1}t_{2}= t_{3}t_{4}= -c/a - Tangents to the parabola y
^{2}= 4ax

1. yy_{1}= 2a(x+x_{1}) at the point (x_{1}, y_{1})

2. y = mx + a/m ( m ≠ 0) at (a/m^{2}, 2a/m)

3. ty = x+at_{2}at (at^{2}, 2at) **Normals to the parabola y**^{2}= 4ax

1. y-y_{1 }= -y_{1}/ 2a(x-x_{1}) at the point (x_{1}, y_{1})

2. y = mx -2am – am^{3}at (am^{2}, -2am)

3. y + tx = 2at +at^{3}at (at^{2}, 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 y
^{2}= 4ax lies on then directrix and has the coordinates –a, a(t_{1 }+ t_{2}+ t_{3}+ t_{1}t_{2}t_{3}). - 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(x
^{2}+ y^{2})– 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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