**Conic Section**

A conic is the locus of a point whose distance from a fixed point bears a constant ratio to its distance from a fixed line. The fixed point is the focus S and the fixed line is directrix l.

The constant ratio is called the eccentricity denoted by e.

- If 0 < e < 1, conic is an ellipse.
- e = 1, conic is a parabola.
- e > 1, conic is a hyperbola.
- If fixed point of curve is (x
_{1}, y_{1}) and fixed line is ax + by + c = then equation of the conic is

(a^{2}+ b^{2}) [(x — x_{1})^{2}+ (y — y_{1})^{2}] = e^{2}(ax + by + c)^{2}

**General Equation of Conic**

A second degree equation ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c= 0 represents

- Pair of straight lines, if
- Circle, if a = b, h = 0
- Parabola, if h
^{2}= ab and Δ ≠ 0 - Ellipse, if h
^{2}< ab and Δ ≠ 0 - Hyperbola, if h
^{2}> ab and Δ ≠ 0 - Rectangular hyperbola, if a + b = 0 and Δ ≠ 0

**Parabola**

A parabola is the locus of a point which moves in a plane such that its distance from a fixed point in the plane is always equal to its distance from a fixed straight line in the same plane.

If focus of a parabola is S(x_{1}, y_{1}) and equation of the directrix is ax + by + c = 0, then the equation of the parabola is

(a^{2} + b^{2})[(x – x_{1})^{2} + (y – y_{1})^{2}] = (ax + by + c)^{2}

**Definitions Related to Parabola**

**Vertex**The intersection point of parabola and axis.**Centre**The point which bisects every chord of the conic passing through it.**Focal Chord**Any chord passing through the focus.**Double Ordinate**A chord perpendicular to the axis of a conic.**Latusrectum**A double ordinate passing through the focus of the parabola.**Focal Distance**The distance of a point P(x, y) from the focus S is called the focal distance of the point P.

**Other Forms of a Parabola**

If the vertex of the parabola is at a point A(h , k) and its latusrectum is of length 4a, then its equation is

- (y – k)
^{2}= 4a (x – h), its axis is parallel to OX i. e. , parabola open rightward. - (y – k)
^{2}= – 4a (x – h), its axis is parallel to OX’ i. e., parabola open leftward. - (x – h)
^{2}= – 4a (y – k), its axis is parallel to OY i.e., parabola open upward. - (x – h)
^{2}= – 4a (y – k), its axis is parallel to OY ‘ i.e., parabola open downward. – - The general equation of a parabola whose axis is parallel to X – axis is x = ay
^{2}+ by + c and the general equation of a parabola whose axis is parallel to Y-axis is y = ax^{2}+ bx + c.

**Position of a Point**

The point (x_{1}, y_{1}) lies outside, on or inside the parabola y^{2} = 4ax according as y_{1}^{2} — 4ax_{1} >, =, < 0.

**Chord**

Joining any two points on a curve is called chord.

(i) Parametric Equation of a Chord Let P(at_{1}^{2} , 2at_{1}) and Q (at_{2}^{2}, 2at_{2}) be any two points on the parabola y_{1} = 4ax, then the equation of the chord is

or y (t_{1} + t_{1}) = 2x + 2at_{1}t_{2}

(ii) Let P(at^{2} , 2at) be the one end of a focal chord PQ of the parabola y^{2} = 4ax, then the coordinates of the other end Q are

(a/t^{2}, -2a/t)

(iii) If l_{1} and l_{2} are the length of the focal segments, then length of the latusrectum = 2 (harmonic mean of focal segment)

i.e.,

(iv) For a chord joining points P(at_{1}^{2} , 2at_{1}) and Q(at_{2}^{2} , 2at_{2}) and passing through focus, then t_{1}t_{2} = 1.

(v) Length of the focal chord having t_{1} and t_{2} as end points is a (t_{1} — t_{1})^{2}.

(vi) Chord of contact drawn from a point (x_{1}, y_{1}) to the parabola y^{2} = 4ax is yy_{1}, = 2a (x + x_{1})

(vii) Equation of the chord of the parabola y^{2} = 4ax, which is bisected at (x_{1} , y_{1}) is given by

T = S_{1}

i.e. , yy_{1} — 2a (x + x_{1}) = y_{1}^{2} – 4ax

**Equation of Tangent**

A line which touch only one point of a parabola.

(i) Point Form The equation of the tangent to the parabola y^{2} = 4ax at a point (x_{1}, y_{1}) is given by yy_{1} = 2a (x + x_{1})

(ii) Slope Form

(a) The equation of the tangent of slope m to the parabola y^{2} = 4ax is

y = (mx + a/m)

(b) The equation of the tangent of slope m to the parabola (y – k)^{2} = 4a (x – h) is given by

(y – k)^{2} = m (x — h) + a/m

The coordinates of the point of contact are

(iii) Parametric Form The equation of the tangent to the parabola y^{2} = 4ax at a point (at^{2}, 2at) is yt = x + at^{2}

(iv) The line y = mx + c touches a parabola, if c = a/m and the point of contact is

(v) Point of Intersection of Two Tangents Let two tangents at P(at_{1}^{2} , 2at_{1}) and Q(at_{2}^{2}, 2at_{2}) intersect at R. Then, their point of intersection is R[at_{1}t_{2}, a(t_{1} + t_{2})] i.e., (GM of abscissa, AM of ordinate).

(vi) The straight line lx + my + n = 0 touches y^{2} = 4ax, if nl = am^{2} and x cos α + y sin α = p touches y^{2} = 4ax, if p cos α + a sin^{2} α = 0.

(vii) Angle θ between tangents at two points P(at_{1}t_{2} , 2at_{1}) and Q(at_{2}^{2}, 2at_{2}) on the parabola y^{2} = 4ax is given by

(viii) The combined equation of the pair of tangents drawn from a point to a parabola y^{2} = 4ax is given by

SS_{1} = T^{2}

where, S = y^{2} – 4ax, S_{1}= y_{1}^{2} – 4ax_{1}

and T = [yy_{1} – 2a (x + x_{1})]

**Important Results on Tangents**

- The tangent at any point on a parabola bisects the angle between the focal distance of the point and the perpendicular on the directrix from the point.
- The tangent at the extremities of a focal chord of a parabola intersect at right angle on the directrix.
- The portion of the tangent to a parabola cut off between the directrix and the curve subtends a right angle at the focus.
- The perpendicular drawn from the focus on any tangent to a parabola intersect it at the point where it cuts the tangent at the vertex.
- The orthocentre of any triangle formed by three tangents to a parabola lies on the directrix.
- The circumcircle formed by the intersection points of tangents at any three points on a parabola passes through the focus of the parabola.
- The tangent at any point of a parabola is equally inclined to the focal distance of the point and the axis of the parabola.
- The length of the subtangent at any point on a parabola is equal to twice the abscissa of the point.
- Two tangents can be drawn from a point to a parabola. Two tangents are real and distinct or coincident or imaginary according as given point lies outside, on or inside the parabola.

**Equation of Normal**

A line which is perpendicular to the tangent.

- Point Form The equation of the normal to the parabola y
^{2}= 4ax at a point (x_{1}, y_{1}) is given by y – y_{1}= -y_{1}/2a(x — x_{1}). - Parametric Form The equation of the normal to the parabola y
^{2}= 4ax at point (at^{2}, 2at) is given by y + tx = 2at + at^{3} - Slope Form The equation of the normal to the parabola y
^{2}= 4ax in terms of its slope m is given by y = mx — 2am — am^{3}at point (am^{2}, — 2am).

**Important Results on Normals**

- If the normal at the point P(at
_{1}^{2},2at_{1}) meets the parabola y^{2}= 4ax at Q(a_{2}^{2},2at_{2}), then t_{2}= -t – 2/t_{1}. - The tangent at one extremity of the focal chord of a parabola is parallel to the normal at other extremity.
- The normal at points P(at
_{1}^{2},2at_{1}) and Q(a_{2}^{2},2at_{2}) to the parabola y^{2}= 4ax intersect at the point

[2a + a(t_{1}^{2}+ t_{2}^{2}+ t_{1}t_{2})], – at_{1}t_{2}(t_{1}+ t_{2}). - If the normal at points P(at
_{1}^{2},2at_{1}) and Q(a_{2}^{2},2at_{2}) on the parabola y^{2}= 4ax meet on the parabola, then t_{1}t_{2}= 2. - If the normal at two points P and Q of a parabola y
^{2}= 4ax intersect at a third point R on the curve, then the product of the ordinates of P and Q is 8a^{2}. - If the normal chord at a point P(at
^{2},2at)to the parabola y^{2}= 4ax subtends a right angle at the vertex of the parabola, then t^{2}= 2. - The normal chord of a parabola at a point whose ordinate is equal to the abscissa, subtends a right angle at the focus.
- The normal at any point of a parabola is equally inclined to the focal distance of the point and the axis of the parabola.
- Three normals can be drawn from a point to a parabola.
- Conormai The points on the parabola at which the normals pass through a common point are called conormal points. The conormal points are called the feet of the normals.

Points A, B and Care called conormal points.

- The algebraic sum of the slopes of the normals at conormals point is O.
- The sum of the ordinates of the conormal points is O.
- The centroid of the triangle formed by the conormal points on a parabola lies on its axis.

**Director Circle**

- The locus of the point of intersection of perpendicular tangents to a conic is known as director circle.
- The director circle of a parabola is its directrix.

**Equation of Diameter**

- The locus of mid-point of a system of parallel chords of a conic is known its diameter.
- The diameter bisecting chords of slope m to the parabola y
^{2}= 4ax is y = 2a/m

**Length of Tangent and Normal**

- The length of the tangent = PT = PN cosec Ψ = y
_{1}cosec Ψ - The length of subtangent = NT = PN cot Ψ = y
_{1}cot Ψ - The length of normal = PG = PN sec Ψ = y
_{1}sec Ψ - The length of subnormal = NG = PN tan Ψ = y
_{1}tan Ψ

**Pole and Polar**

Let P be a point lying within or outside a given parabola. Suppose any straight line drawn through P intersects the parabola at Q and R. Then, the locus of the point of intersection of the tangents to the parabola at Q and R is called the polar of the given point P with respect to the parabola and the point P is called the pole of the polar.

- The polar of a point P(x
_{1},_{1}) with respect to the parabola y^{2}= 4ax is yy_{1}= 2a(x + x_{1}) or T = 0. - Any tangent is the polar of its point of contact.
- Pole of lx + my + n = 0 with respect to y
^{2}= 4ax is - Pole of the chord joining (x
_{1}, y_{1}) and (x_{1}, y_{1}) is - If the polar of P(x
_{1}, y_{1})passes through Q(x_{2},y_{2}), then the polar of Q will passes through P. Here, P and Q are called conjugate points. - If the pole of a line a
_{1}x + b_{1}y + c_{1}= 0 lies on another line a_{2}x + b_{2}y + c_{2}= 0, then the pole of the second line will lies on the first line. Such lines are called conjugate lines. - The point of intersection of the polar of two points Q and R is the pole of QR.
- The tangents at the ends of any chord of the parabola meet on the diameter which bisect the chord.

**Important Points to be Remembered**

(i) For the ends of latusrectum of the parabola y^{2} = 4ax, the values of the perimeter are ± 1.

(ii) The circles described on focal radii of a parabola as diameter touches the tangent at the vertex.

(iii) The straight line y =mx + c meets the parabola y^{2} = 4ax in two points. These two points are real and distinct, if c > a/m, points are real and coincident, if c = a/m, points are imaginary, if c < a/m.

(iv) 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.

(v) The circles described on any focal chord of a parabola as diameter touches the directrix.

(vi) If y_{1}, y_{2}, y_{3} are the ordinates of the vertices of a triangle inscribed in the parabola y^{2} = 4ax, then its area is 1/8a (y_{1} – y_{2}) (y_{2} – y_{3}) (y_{3} – y_{1}).

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