Tangent to a parabola Assignment Help

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Tangent to a parabola:

Tangent at the Point (x1, y1):

Let equation of the parabola be y2 = 4ax.

Hence, the value of dy/dx at P(x1, y1) is 2a/y1 and the equation of tangent at P is

y - y1 =2a/y1 (x - x1)   i.e. yy1 = 2a(x - x1) + y12 => yy1 = 2a(x + x1)

Tangent in Terms of m:

Assume that the equation of a tangent to parabola y2 = 4ax             ......(1)

      is y = mx + c                                                                                 ......(2)

The abscissae of points of intersection of (1) and (2) can be given by the equation
(mx + c)2 = 4ax. But condition that the straight line (ii) should touch parabola is that it should meet parabola in coincident points

      => (mc - 2a)2 = m2c2                                                                    ......(3)

      =>  c = a /m.

Thus, y = mx + a/m is a tangent to the parabola y2 = 4ax, whatever be the m.

Equation (mx +c)2 = 4ax  now becomes (mx - a/m)2  = 0

      => x =  and y2 = 4ax => y = 2a/m.

Therefore the point of contact of tangent y = mx + a/m is (a/m2, 2a/m).

Tangent at the Point 't':

Let the equation of parabola is y2 = 4ax.

The equation of tangent at (x1, y1) to this parabola is yy1 = 2a(x + x1).

If point (x1, y1) ≡ (at2, 2at)

Equation of the tangent becomes y.2at = 2a(x + at2) => yt = x + at2.

Note:

  • The point of intersection of tangents at 't1' and 't2' to parabola y2 = 4ax is
    (at1t2, a(t1 + t2)).

Example: Two tangents are drawn from point (-2, -1) to parabola y2 = 4x. If a is the angle between these tangents, then tan a equals  

                        (A) 3                                                   (B) 1/3

                        (C) 2                                                   (D) 1/2

Solution:        Here a = 1. Any tangent is y = mx + 1/m.

                        It passes through (-2, -1)

                       ∴ 2m2 -m -1 = 0

                        2354_Tangent to a parabola.png

                        Hence (A) is the required answer.      

 

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