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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 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 are
1. 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)
7. The ends of the double ordinate of the parabola can be taken as (at^{2}, 2at) and (at^{2},-2at).
The parabola y = a(x – h)^{2} + k has its vertex at (h, k)
The perpendicular distance from focus on directrix is half the length of latus rectum
For the parabola y = Ax^{2} + Bx + C, the length of the latus rectum is 1/|A| and axis is parallel to y-axis. If A is positive, then it is concave up parabola and if A is negative then it is concave down parabola.
For the parabola x = Ay^{2} + By + C, the length of the latus rectum is 1/|A| and axis is parallel to x-axis. If A is positive, then it is opening right parabola and if A is negative then it is opening left parabola.
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
The length of the focal chord having parameters t_{1} and t_{2} for its end points is a(t_{2 }- t_{1})^{2}.
The length of the smallest focal chord of the parabola is 4a, which is the latus rectum of the parabola.
Tangents to the parabola y^{2} = 4ax
Point Form:
yy_{1} = 2a(x+x_{1}) at the point (x_{1}, y_{1})
Slope Form:
y = mx + a/m ( m ≠ 0) at (a/m^{2}, 2a/m)
Parametric Form:
ty = x + at^{2} at point (at^{2}, 2at)
The coordinates of points of intersection of the tangents at two points P(at_{1}^{2}, 2at_{1}) and Q(at_{2}^{2}, 2at_{2}) are S(at_{1}t_{2}, a(t_{1} + t_{2})).
?The arithmetic mean of y coordinate of P and Q is the y-coordinate of the point of intersection of tangents at P and Q on the parabola.
The geometric mean of the x-coordinate of P and Q is the x-coordinate of the point of intersection of tangents at P and Q on the parabola.
Normal to the parabola y^{2} = 4ax
Point form
y-y_{1} = -y_{1}/2a(x-x_{1}) at the point (x_{1}, y_{1})
Slope Form
y = mx - 2am – am^{3} at (am^{2}, -2am)
Parametric Form
y + tx = 2at + at^{3} at (at^{2}, 2at)
The point of intersection of normals at any two points say P(at_{1}^{2}, 2at_{1}) and Q(at_{2}^{2}, 2at_{2}) on the parabola is given by R(2a + a(t_{1}^{2} + t_{2}^{2} + t_{1}t_{2}), -at_{1}t_{2}(t_{1 }+ t_{2})).
The algebraic sum of the slopes of three concurrent normals is zero.
The algebraic sum of ordinates of the feet of three normals drawn to a parabola form a given point is zero.
The centroid of the triangle formed by the feet of three normals lies on the axis of the parabola.
The equation of the chord of the parabola y^{2} = 4ax whose middle point is P(x_{1},y_{1}) is
yy_{1} – 2a(x – x_{1}) = y_{1}^{2} – 4ax_{1}.
The equation of pair of tangents from the point P(x_{1}, y_{1}) to the parabola y^{2} = 4ax is given by
[yy_{1} – 2a(x + x_{1})]^{2} = (y^{2} – 4ax)(y_{1}^{2} – 4ax_{1}).
The equation of the polar of the point P(x_{1},y_{1}) to the parabola y^{2} = 4ax is y_{1} = 2a(x + x_{1}).
The pole of the line lx + my + n = 0 to the parabola y^{2} = 4ax is (n/l, -2am/l).
The polar of the focus of the parabola is the directrix.
Two straight lines are said to be conjugated to each other with respect to a parabola when the pole of one lies on the other. Similarly, two points P and Q are said to be conjugate points if polar of P passes through Q and vice versa.
The equation of the diameter of the parabola is y = 2a/m which is parallel to its axis.
The length of the tangent to the parabola y^{2} = 4ax which makes an angle φ with the x - axis is y cosec φ
The length of normal of the parabola is y sec φ, where φ is the angle made by the tangent with the x - axis.
The length of sub tangent is y cot φ, where φ is the angle made by the tangent with the x - axis.
The length of sub normal is y cot φ, where φ is the angle made by the tangent with the x - axis.
The foot of the perpendicular from the focus on any tangent to a parabola lies on the tangent at the vertex.
The tangents at the extremities of the focal chord intersect at right angles on the directrix.
Circle described on the focal length as diameter touches the tangent at the vertex.
Circle described on the focal chord as diameter touches the directrix.
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 orthocentre 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 vital parabolas along with their basic components like vertex and directrix are tabulated below:
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