Conic Sections
Exercise 10.3 • Ellipses
Ellipse Parameters ($a > b$):
Relation: $c^2 = a^2 – b^2$, Eccentricity $e = c/a$, Latus Rectum = $2b^2/a$
Horizontal ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$): Foci $(\pm c, 0)$, Vertices $(\pm a, 0)$
Vertical ($\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$): Foci $(0, \pm c)$, Vertices $(0, \pm a)$
Relation: $c^2 = a^2 – b^2$, Eccentricity $e = c/a$, Latus Rectum = $2b^2/a$
Horizontal ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$): Foci $(\pm c, 0)$, Vertices $(\pm a, 0)$
Vertical ($\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$): Foci $(0, \pm c)$, Vertices $(0, \pm a)$
Q1
1. $\frac{x^2}{36} + \frac{y^2}{16} = 1$
Identify
Denominator of $x^2$ is larger ($36 > 16$). Horizontal Ellipse.
Values
$a^2 = 36 \Rightarrow a = 6$. $b^2 = 16 \Rightarrow b = 4$.
Find c
$c = \sqrt{a^2 – b^2} = \sqrt{36 – 16} = \sqrt{20} = 2\sqrt{5}$.
Calc
Foci: $(\pm 2\sqrt{5}, 0)$, Vertices: $(\pm 6, 0)$
Major Axis: $2a=12$, Minor Axis: $2b=8$
$e = \frac{c}{a} = \frac{2\sqrt{5}}{6} = \frac{\sqrt{5}}{3}$, L.R. = $\frac{2b^2}{a} = \frac{2(16)}{6} = \frac{16}{3}$
Major Axis: $2a=12$, Minor Axis: $2b=8$
$e = \frac{c}{a} = \frac{2\sqrt{5}}{6} = \frac{\sqrt{5}}{3}$, L.R. = $\frac{2b^2}{a} = \frac{2(16)}{6} = \frac{16}{3}$
Foci $(\pm 2\sqrt{5}, 0)$, Vertices $(\pm 6, 0)$
Q2
2. $\frac{x^2}{4} + \frac{y^2}{25} = 1$
Identify
Denominator of $y^2$ is larger ($25 > 4$). Vertical Ellipse.
Values
$a^2 = 25 \Rightarrow a = 5$. $b^2 = 4 \Rightarrow b = 2$.
Find c
$c = \sqrt{25 – 4} = \sqrt{21}$.
Calc
Foci: $(0, \pm \sqrt{21})$, Vertices: $(0, \pm 5)$
Major: 10, Minor: 4
$e = \frac{\sqrt{21}}{5}$, L.R. = $\frac{2(4)}{5} = \frac{8}{5}$
Major: 10, Minor: 4
$e = \frac{\sqrt{21}}{5}$, L.R. = $\frac{2(4)}{5} = \frac{8}{5}$
Foci $(0, \pm \sqrt{21})$, Vertices $(0, \pm 5)$
Q3
3. $\frac{x^2}{16} + \frac{y^2}{9} = 1$
Identify
Horizontal ($16 > 9$). $a=4, b=3$.
c
$c = \sqrt{16-9} = \sqrt{7}$.
Foci $(\pm \sqrt{7}, 0)$, Vertices $(\pm 4, 0)$, $e=\frac{\sqrt{7}}{4}$, LR=$\frac{9}{2}$
Q4
4. $\frac{x^2}{25} + \frac{y^2}{100} = 1$
Identify
Vertical ($100 > 25$). $a=10, b=5$.
c
$c = \sqrt{100-25} = \sqrt{75} = 5\sqrt{3}$.
Foci $(0, \pm 5\sqrt{3})$, Vertices $(0, \pm 10)$, $e=\frac{\sqrt{3}}{2}$, LR=5
Q5
5. $\frac{x^2}{49} + \frac{y^2}{36} = 1$
Identify
Horizontal ($49 > 36$). $a=7, b=6$.
c
$c = \sqrt{49-36} = \sqrt{13}$.
Foci $(\pm \sqrt{13}, 0)$, Vertices $(\pm 7, 0)$, $e=\frac{\sqrt{13}}{7}$, LR=$\frac{72}{7}$
Q6
6. $\frac{x^2}{100} + \frac{y^2}{400} = 1$
Identify
Vertical ($400 > 100$). $a=20, b=10$.
c
$c = \sqrt{400-100} = \sqrt{300} = 10\sqrt{3}$.
Foci $(0, \pm 10\sqrt{3})$, Vertices $(0, \pm 20)$, $e=\frac{\sqrt{3}}{2}$, LR=10
Q7
7. $36x^2 + 4y^2 = 144$
Rewrite
Divide by 144: $\frac{36x^2}{144} + \frac{4y^2}{144} = 1 \Rightarrow \frac{x^2}{4} + \frac{y^2}{36} = 1$.
Identify
Vertical ($36 > 4$). $a=6, b=2$.
c
$c = \sqrt{36-4} = \sqrt{32} = 4\sqrt{2}$.
Foci $(0, \pm 4\sqrt{2})$, Vertices $(0, \pm 6)$, $e=\frac{2\sqrt{2}}{3}$, LR=$\frac{4}{3}$
Q8
8. $16x^2 + y^2 = 16$
Rewrite
Divide by 16: $\frac{x^2}{1} + \frac{y^2}{16} = 1$.
Identify
Vertical ($16 > 1$). $a=4, b=1$.
c
$c = \sqrt{16-1} = \sqrt{15}$.
Foci $(0, \pm \sqrt{15})$, Vertices $(0, \pm 4)$, $e=\frac{\sqrt{15}}{4}$, LR=$\frac{1}{2}$
Q9
9. $4x^2 + 9y^2 = 36$
Rewrite
Divide by 36: $\frac{x^2}{9} + \frac{y^2}{4} = 1$.
Identify
Horizontal ($9 > 4$). $a=3, b=2$.
c
$c = \sqrt{9-4} = \sqrt{5}$.
Foci $(\pm \sqrt{5}, 0)$, Vertices $(\pm 3, 0)$, $e=\frac{\sqrt{5}}{3}$, LR=$\frac{8}{3}$
Q10
10. Vertices $(\pm 5, 0)$, Foci $(\pm 4, 0)$
Type
Vertices on x-axis $\Rightarrow$ Horizontal. $a=5$.
Find b
$c=4$. $b^2 = a^2 – c^2 = 25 – 16 = 9$.
$\frac{x^2}{25} + \frac{y^2}{9} = 1$
Q11
11. Vertices $(0, \pm 13)$, Foci $(0, \pm 5)$
Type
Vertices on y-axis $\Rightarrow$ Vertical. $a=13$.
Find b
$c=5$. $b^2 = 13^2 – 5^2 = 169 – 25 = 144$.
$\frac{x^2}{144} + \frac{y^2}{169} = 1$
Q12
12. Vertices $(\pm 6, 0)$, Foci $(\pm 4, 0)$
Type
Horizontal. $a=6, c=4$.
Find b
$b^2 = 36 – 16 = 20$.
$\frac{x^2}{36} + \frac{y^2}{20} = 1$
Q13
13. Ends of major axis $(\pm 3, 0)$, ends of minor axis $(0, \pm 2)$
Values
Major on x-axis $\Rightarrow a=3$. Minor $\Rightarrow b=2$.
$\frac{x^2}{9} + \frac{y^2}{4} = 1$
Q14
14. Ends of major axis $(0, \pm \sqrt{5})$, ends of minor axis $(\pm 1, 0)$
Values
Major on y-axis $\Rightarrow a=\sqrt{5} \Rightarrow a^2=5$. Minor $\Rightarrow b=1$.
$\frac{x^2}{1} + \frac{y^2}{5} = 1$
Q15
15. Length of major axis 26, Foci $(\pm 5, 0)$
Values
$2a=26 \Rightarrow a=13$. Foci on x-axis $\Rightarrow c=5$.
Find b
$b^2 = 13^2 – 5^2 = 144$.
$\frac{x^2}{169} + \frac{y^2}{144} = 1$
Q16
16. Length of minor axis 16, Foci $(0, \pm 6)$
Values
$2b=16 \Rightarrow b=8$. Foci on y-axis $\Rightarrow$ Vertical. $c=6$.
Find a
$a^2 = b^2 + c^2 = 64 + 36 = 100$.
$\frac{x^2}{64} + \frac{y^2}{100} = 1$
Q17
17. Foci $(\pm 3, 0), a = 4$
Values
$c=3, a=4$. Horizontal.
Find b
$b^2 = 16 – 9 = 7$.
$\frac{x^2}{16} + \frac{y^2}{7} = 1$
Q18
18. $b = 3, c = 4$, centre origin, foci on x-axis
Find a
$a^2 = b^2 + c^2 = 9 + 16 = 25$.
$\frac{x^2}{25} + \frac{y^2}{9} = 1$
Q19
19. Centre (0,0), major axis y-axis, passes through (3, 2) and (1, 6).
Form
$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$.
Points
$(3,2) \Rightarrow \frac{9}{b^2} + \frac{4}{a^2} = 1 \dots (1)$
$(1,6) \Rightarrow \frac{1}{b^2} + \frac{36}{a^2} = 1 \dots (2)$
$(1,6) \Rightarrow \frac{1}{b^2} + \frac{36}{a^2} = 1 \dots (2)$
Solve
Multiply (2) by 9: $\frac{9}{b^2} + \frac{324}{a^2} = 9$. Subtract (1): $\frac{320}{a^2} = 8 \Rightarrow a^2=40$.
Sub into (2): $\frac{1}{b^2} + \frac{36}{40} = 1 \Rightarrow \frac{1}{b^2} = 1 – 0.9 = 0.1 \Rightarrow b^2 = 10$.
Sub into (2): $\frac{1}{b^2} + \frac{36}{40} = 1 \Rightarrow \frac{1}{b^2} = 1 – 0.9 = 0.1 \Rightarrow b^2 = 10$.
$\frac{x^2}{10} + \frac{y^2}{40} = 1$
Q20
20. Major axis x-axis, passes through (4, 3) and (6, 2).
Form
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
Points
$(4,3) \Rightarrow \frac{16}{a^2} + \frac{9}{b^2} = 1 \dots (1)$
$(6,2) \Rightarrow \frac{36}{a^2} + \frac{4}{b^2} = 1 \dots (2)$
$(6,2) \Rightarrow \frac{36}{a^2} + \frac{4}{b^2} = 1 \dots (2)$
Solve
Solving system yields $a^2 = 52, b^2 = 13$.
$\frac{x^2}{52} + \frac{y^2}{13} = 1$