# FIND THE EQUATION OF THE TANGENT TO THE PARABOLA IN PARAMETRIC FORM

Find the Equation of the Tangent to the Parabola in Parametric Form :

Here we are going to see some practice questions to find the equation of the tangent to the parabola in parametric form.

 Name of the conicCircleParabolaEllipseHyperbola Parametric equationsx = a cos θ , y = a sin θx = at2 , y = 2atx = a cos θ , y = b sin θx = a sec θ , y = b tan θ

## Find the Equation of the Tangent to the Parabola in Parametric Form - Practice questions

Question 1 :

Find the equation of the tangent at t = 2 to the parabola y2 = 8x . (Hint: use parametric form)

Solution :

x = at2, y = 2at

y2 = 8x

4a  =  8  ==>  a  =  2

x = 2(2)2  ==>  8

y = 2(2)(2)  ==> 8

The point on the tangent line is (8, 8).

Equation of tangent to the parabola :

yy1 = 8[(x + x1)/2]

y(8) = 4(x + 8)

8y  =  4x + 32

4x - 8y + 32  =  0

Divide the equation by 4, we get

x - 2y + 8  =  0

Hence the required equation of the tangent line is x - 2y + 8  =  0.

Question 2 :

Find the equations of the tangent and normal to hyperbola 12x2 − 9y2 = 108 at θ  =  π/3 . (Hint : use parametric form)

Solution :

x = a sec θ , y = b tan θ

(12x2/108) − (9y2/108) = 108/108

(12x2/108) − (9y2/108) = 108/108

(x2/9) − (y2/12) = 1

a2  =  9, b2  =  12

a  =  3 and b  =  2√3

 x = a sec θx  =  3 sec π/3x  =  3(2)  =  6 y = b tan θy  =  2√3 tan π/3y  =  2√3(√3)  =  6

Equation of tangent :

12(xx1) − 9(yy1)  =  108

12x(6) − 9y (6)  =  108

72x - 54y  =  108

8x - 6y  =  12

4x - 3y - 4  =  0

Equation of normal :

3x + 4y + k  =  0

3(6) + 4(6) + k  =  0

18 + 24 + k  =  0

k  =  -42

3x + 4y - 42  =  0

Prove that the point of intersection of the tangents at ‘ t1 ’ and ‘ t2 ’on the parabola y2 = 4ax is a t1 t2 , a(t1 + t2)

If the normal at the point ‘ t1 ’ on the parabola y2 = 4ax meets the parabola again at the point ‘ t2 ’, then prove that t2  =  -(t1 + 2/t1)

After having gone through the stuff given above, we hope that the students would have understood, "Find the Equation of the Tangent to the Parabola in Parametric Form".

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