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Revision Notes on Parabola
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 parabolax^{2} = 4ay opens upwards.
The equation y^{2} = 4ax is considered to be the standard equation of the parabola for which the various components are
Vertex at (0,0)
Directrix is x+a = 0
Axis is y = 0
Focus is (a, 0)
Length of latus rectum = 4a
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 parabolay^{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
yy_{1} = 2a(x+x_{1}) at the point (x_{1}, y_{1})
y = mx + a/m ( m ≠ 0) at (a/m^{2}, 2a/m)
ty = x+at^{2} at (at^{2}, 2at)
Normals to the parabola y^{2} = 4ax
y-y_{1} = -y_{1}/ 2a(x-x_{1}) at the point (x_{1}, y_{1})
y = mx -2am – am^{3} at (am^{2}, -2am)
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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