Parametric Equations of a Parabola:
If coordinates of any point (x, y) on a curve can be expressed as functions of a variable t, which are given by x = f (t), y = y (t) ......(1)
Then equations in (1) are said to be parametric equations of curve where't' is called as parameter.
Clearly x = at2, y = 2at satisfy the equation y2 = 4ax for all the real values of t. Hence parametric equations of parabola y2 = 4ax are x = at2, y = 2at, where t is parameter.
Also, (at2, 2at) is a point on parabola y2 = 4ax for all the real values of t. This point is described as the point 't' on parabola.
Example: The tangents at points (at12, 2at1), (at22, 2at2) on parabola y2 = 4ax are at right angles if
(A) t1t2 = -1 (B) t1t2 = 1
(C) t1t2 = 2 (D) t1t2 = -2
Solution: 1/t1.1/t2 = -1 => t1t2 = -1
Hence (A) is the required answer.
Example: If P(at12, 2at1) and Q (at22, 2at2) are 2 variable points on the curve y2 = 4ax and PQ subtends the right angle at vertex, then t1t2 is equal to
(A) -1 (B) -2
(C) -3 (D) -4
Solution: Slope of OP = 2/t1
Slope of OQ = 2/t1
Given OP ⊥ OQ =>2/t1.2/t2 = -1 => t1t2 = -4
Hence (D) is the reuired answer.
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