Equations of tangent in different forms

Hyperbola

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Hyperbola

A hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed line is always constant which is always greater than unity.
Hyperbola 1

Standard equation of the hyperbola

Let S be the focus, ZM be the directrix and e be the eccentricity of the hyperbola, then by definition, Hyperbola 2, where b2 = a2(e2 − 1).
Hyperbola 3

Conjugate hyperbola

The hyperbola whose transverse and conjugate axis are respectively the conjugate and transverse axis of a given hyperbola is called conjugate hyperbola of the given hyperbola.
Hyperbola 4Difference between both hyperbolas will be clear from the following table:
Hyperbola 5

Special form of hyperbola

If the centre of hyperbola is (h, k) and axes are parallel to the co-ordinate axes, then its equation is Hyperbola 6.

Auxiliary circle of hyperbola

Let Hyperbola 7 be the hyperbola, then equation of the auxiliary circle is x2 + y2 = a2.
Let ∠QCN = ϕ. Here P and Q are the corresponding points on the hyperbola and the auxiliary circle (0 ≤ ϕ < 2π).
Hyperbola 8

Parametric equations of hyperbola

The equations x = a sec ϕ and y = b tan ϕ are known as the parametric equations of the hyperbola Hyperbola 9.
This (a sec ϕ, b tan ϕ) lies on the hyperbola for all values of ϕ.

Position of a point with respect to a hyperbola

Hyperbola 10

Intersection of a line and a hyperbola

Hyperbola 11

Equations of tangent in different forms

Hyperbola 12

Ellipse

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Ellipse

An ellipse is the locus of a point which moves in such a way that its distance from a fixed point is in constant ratio (<1) to its distance from a fixed line. The fixed point is called the focus and fixed line is called the directrix and the constant ratio is called the eccentricity of the ellipse, denoted by (e).
Ellipse 1

Standard equation of the ellipse

Let S be the focus, ZM be the directrix of the ellipse and P(x, y) is any point on the ellipse, then by definition Ellipse 2, where b2 = a2(1 – e2).
Since e < 1, therefore a2(1 – e2) < a2 ⇒ b2 < a2.
Ellipse 3 The other form of equation of ellipse is Ellipse 4, where, a2 = b2(1 – e2) i.e., a < b.
Ellipse 5Difference between both ellipses will be clear from the following table:
Ellipse 6

Parametric form of the ellipse

Let the equation of ellipse in standard form will be given by Ellipse 7.
Then the equation of ellipse in the parametric form will be given by x = a cos ϕ, y = b sin ϕ, where ϕ is the eccentric angle whose value vary from 0 ≤ ϕ < 2π. Therefore coordinate of any point P on the ellipse will be given by (a cos ϕ, b sin ϕ).

Special forms of an ellipse

(1) If the centre of the ellipse is at point and the directions of the axes are parallel to the coordinate axes, then its equation is
Ellipse 8
Ellipse 9

Position of a point with respect to an ellipse

Ellipse 10

Intersection of a line and an ellipse

The line y = mx + c intersects the ellipse Ellipse 11 in two distinct points if a2m2 + b2 > c2, in one point if c2 = a2m2 + b2 and does not intersect if a2m2 + b2 < c2.

Equations of tangent in different forms

Ellipse 12

Parabola

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Parabola

A parabola is the locus of a point which moves in a plane such that its distance from a fixed point (i.e., focus) in the plane is always equal to its distance from a fixed straight line (i.e., directrix) in the same plane.
Parabola 1

Standard equation of the parabola

Let S be the focus, ZZ‘ be the directrix of the parabola and be any point on parabola, then standard form of the parabola is y2 = 4ax.
Some other standard forms of parabola are

  1. Parabola opening to left i.e, y2 = –4ax.
  2. Parabola opening upwards i.e., x2 = 4ay.
  3. Parabola opening downwards i.e., x2 = –4ay.

Some terms related to parabola

Parabola 2

Important termsy2 = 4axy2 = –4axx2 = 4ayx2 = –4ay
Vertex(0, 0)(0, 0)(0, 0)(0, 0)
Focus(a, 0)(–a, 0)(0, a)(0, –a)
Directrixx = –ax = ay = –ay = a
Axisy = 0y = 0x = 0x = 0
Latusrectum4a4a4a4a
Focal distance P(x, y)x + aa – xy + aa – y

Special form of parabola (y – k)2  = 4a(x – h) = a

The equation of a parabola with its vertex at (h, k) and axis as parallel to x-axis is (y – k)2  = 4a(x – h).
Parabola 3If the vertex of the parabola is (p, q) and its axis is parallel to y-axis, then the equation of the parabola is (x – p)2  = 4b(y – q).

Parametric equations of a parabola

Parabolay2 = 4axy2 = –4axx2 = 4ayx2 = –4ay
Parametric Co-ordinates(at2, 2at)(–at2, 2at)(2at , at2)(2at , –at2)
Parametric Equationsx = at2
y = 2at		x = –at2
y = 2at		x = 2at
y = at2		x = 2at
y = –at2

The parametric equations of parabola (y – k)2  = 4a(x – h) are x = h + at2 and y = k + 2at.

Position of a point and a line with respect to a parabola

(1) Position of a point with respect to a parabola:
The point P(x1, y1) lies outside, on or inside the parabola y2 = 4ax according as y12 = 4ax1 >, =, < 0.
Parabola 4(2) Intersection of a line and a parabola:
The line y = mx + c does not intersect, touches or intersect a parabola y2 = 4ax, according as >, =, < a/m.

Condition of tangency: The line touches the parabola, if c = a/m.

Equations of tangent in different forms

(1) Point Form

Equations of tangent of all other standard parabolas at (x1, y1)
Equation of parabolaTangent at (x1, y1)
y2 = 4axyy1 = 2a (x + x1)
y2 = –4axyy1 = –2a (x + x1)
x2 = 4ayxx1 = 2a (y + y1)
x2 = –4ayxx1 = –2a (y + y1)

(2) Parametric form

Equations of tangent of all other standard parabolas at ‘t’
Equations of parabolasParametric coordinates ‘t’Tangent at ‘t’
y2 = 4ax(at2, 2at)ty = x + at2
y2 = –4ax(–at2, 2at)ty = –x + at2
x2 = 4ay(2at , at2)tx = y + at2
x2 = –4ay(2at , –at2)tx = –y + at2

(3) Slope Form
Parabola 5

Point of intersection of tangents at any two points on the parabola

  1. The point of intersection of tangents at two points P(a1t2, 2at1) and Q(a2t2, 2at2) on the parabola y2 = 4ax is (at1t2, a(t1 + t2)).
    Parabola 6
  2. The locus of the point of intersection of tangents to the parabola y2 = 4ax which meet at an angle α is (x + a)2 tan2 α = y2 – 4ax.
  3. Director circle: The locus of the point of intersection of perpendicular tangents to a conic is known as its director circle. The director circle of a parabola y2 = 4ax is its directrix.
  4. The tangents to the parabola y2 = 4ax at P(a1t2, 2at1) and Q(a2t2, 2at2) intersect at R. Then the area of triangle PQR is 1/2 a2(t1 – t2)3.

Equation of pair of tangents from a point to a parabola

The combined equation of the pair of the tangents drawn from a point to a parabola is SS’ = T2, where S = y2 – 4ax; S’ = y12 – 4ax1 and T = yy1 – 2a(x + x1)
Parabola 7The two tangents can be drawn from a point to a parabola. The two tangent are real and distinct or coincident or imaginary according as the given point lies outside, on or inside the parabola.

Equations of normal in different forms

(1) Point form
Parabola 8
(2) Parametric form
Parabola 9
(3) Slope form
Parabola 10