---
title: "Parametric circle and ellipse with points"
subject: "Computer Aided Design"
chapter: 3
key: "semester_4/computer_aided_design:circle-ellipse-parametric"
source: https://www.bctengineer.com/q/semester_4/computer_aided_design/circle-ellipse-parametric
site: BCT Engineering
---

# Parametric circle and ellipse with points

Computer Aided Design · Chapter 3: Mathematical Representation of Curves

- Asked in: year not given, 6 marks
- Video: [S 2.29 Numerical 1 on Parametric Equation of Ellipse](https://www.youtube.com/watch?v=FSXOqc6z_Dg) — Dr. Khizar Ahmed Pathan
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/computer_aided_design/circle-ellipse-parametric
- Whole chapter: https://www.bctengineer.com/md/semester_4/computer_aided_design/chapter-3

## Question

Write the parametric equations of (a) a circle of centre (2, 3) and radius 5, and (b) an ellipse of centre (1, 2) with semi-major axis 6 along X and semi-minor axis 3. Calculate points on the circle at 0, 60, 120 and 180 degrees, the tangent direction at 60 degrees, and the points of the ellipse at 0, 45 and 90 degrees.

## Short answer

Circle: x=2+5cosθ, y=3+5sinθ; points (7,3), (4.5,7.33), (-0.5,7.33), (-3,3); tangent at 60° (-4.33,2.5). Ellipse: x=1+6cosθ, y=2+3sinθ; points (7,2), (5.243,4.121), (1,5).

## Answer

#### (a) Circle
$$
x=x_c+R\cos\theta,\qquad y=y_c+R\sin\theta,\qquad 0\le\theta\le2\pi
$$

With $(x_c,y_c)=(2,3)$, $R=5$: $x=2+5\cos\theta$, $y=3+5\sin\theta$.

| $\theta$ | $x$ | $y$ |
|---|---|---|
| 0 deg | 7.000 | 3.000 |
| 60 deg | 4.500 | 7.330 |
| 120 deg | -0.500 | 7.330 |
| 180 deg | -3.000 | 3.000 |

Tangent: $\dfrac{dx}{d\theta}=-5\sin\theta,\ \dfrac{dy}{d\theta}=5\cos\theta$. At $60^\circ$: $(-4.330,\ 2.500)$. Its length is 5, and it is perpendicular to the radius vector $(2.5,\ 4.330)$, since the dot product is $-10.83+10.83=0$.

#### (b) Ellipse
$$
x=x_c+a\cos\theta,\qquad y=y_c+b\sin\theta
$$

With $a=6$, $b=3$, centre $(1,2)$: $x=1+6\cos\theta$, $y=2+3\sin\theta$.

| $\theta$ | $x$ | $y$ |
|---|---|---|
| 0 deg | 7.000 | 2.000 |
| 45 deg | 5.243 | 4.121 |
| 90 deg | 1.000 | 5.000 |

Check at 45 deg: $\dfrac{(5.243-1)^2}{36}+\dfrac{(4.121-2)^2}{9}=0.5+0.5=1$.

The circle is the special case $a=b=R$. Note that $\theta$ here is a parameter (the eccentric angle), not the polar angle of the point on the ellipse.

**Answer: circle points (7, 3), (4.5, 7.33), (-0.5, 7.33), (-3, 3); tangent at 60 deg = (-4.33, 2.5); ellipse points (7, 2), (5.243, 4.121), (1, 5).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What are the equations of the circle with centre (2,3), R = 5?
   A. x = 5 + 2cos θ, y = 5 + 3sin θ
   B. x = 2cos θ, y = 3sin θ
   C. x = 2 + 5sin θ, y = 3 + 5sin θ
   D. x = 2 + 5cos θ, y = 3 + 5sin θ
   Answer: D. x = xc + R cos θ, y = yc + R sin θ. (Key point: Circle parametric)

2. What is the tangent direction at 60° on the circle?
   A. (-4.330, 2.500)
   B. (4.330, 2.500)
   C. (2.5, 4.330)
   D. (-2.5, 4.330)
   Answer: A. (-5 sin 60°, 5 cos 60°) = (-4.330, 2.5), perpendicular to the radius. (Key point: Tangent)

3. What is the ellipse with centre (1,2), a = 6, b = 3?
   A. x = 1 + 3cos θ, y = 2 + 6sin θ
   B. x = 1 + 6cos θ, y = 2 + 3sin θ
   C. x = 6 + cos θ, y = 3 + sin θ
   D. x = 6cos θ, y = 3sin θ
   Answer: B. The semi-major axis 6 is along X, so x = 1 + 6 cos θ, y = 2 + 3 sin θ. (Key point: Ellipse)

4. What kind of angle is θ in the ellipse equations?
   A. The polar angle of the point
   B. The tangent angle
   C. A parameter (eccentric angle)
   D. Always equal to the polar angle
   Answer: C. θ is a parameter, not the polar angle of the point on the ellipse. (Key point: Eccentric angle)

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