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 conic

Circle

Parabola

Ellipse

Hyperbola

Parametric equations

x = a cos θ , y = a sin θ

x = at2y = 2at

x = a cos θ , y = b sin θ

x = 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 θ

x  =  3 sec π/3

x  =  3(2)  =  6

y = b tan θ

y  =  2√3 tan π/3

y  =  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".

Apart from the stuff given in this section if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. Problems on Finding Derivative of a Function

    Mar 29, 24 12:11 AM

    Problems on Finding Derivative of a Function

    Read More

  2. How to Solve Age Problems with Ratio

    Mar 28, 24 02:01 AM

    How to Solve Age Problems with Ratio

    Read More

  3. AP Calculus BC Integration of Rational Functions by Partical Fractions

    Mar 26, 24 11:25 PM

    AP Calculus BC Integration of Rational Functions by Partical Fractions (Part - 1)

    Read More