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
Hence (A) is the required answer.
Email based Tangent to a parabola Assignment Help -Tangent to a parabola Homework Help
We at www.expertsmind.com offer email based Tangent to a parabola assignment help - homework help and projects assistance from k-12 school level to university and college level and engineering and management studies. We provide finest service of Mathematics assignment help and Mathematic homework help. Our experts are helping students in their studies and they offer instantaneous tutoring assistance giving their best practiced knowledge and spreading their world class education services through e-Learning program.
Expertsmind's best education services
- Quality assignment-homework help assistance 24x7 hrs
- Best qualified tutor's network
- Time on delivery
- Quality assurance before delivery
- 100% originality and fresh work