Conic Sections
Exercise 10.2 • Parabolas
Standard Parabolas ($a>0$):
1. Right ($y^2=4ax$): Focus $(a,0)$, Dir $x=-a$
2. Left ($y^2=-4ax$): Focus $(-a,0)$, Dir $x=a$
3. Up ($x^2=4ay$): Focus $(0,a)$, Dir $y=-a$
4. Down ($x^2=-4ay$): Focus $(0,-a)$, Dir $y=a$
Length of Latus Rectum = $4a$
1. Right ($y^2=4ax$): Focus $(a,0)$, Dir $x=-a$
2. Left ($y^2=-4ax$): Focus $(-a,0)$, Dir $x=a$
3. Up ($x^2=4ay$): Focus $(0,a)$, Dir $y=-a$
4. Down ($x^2=-4ay$): Focus $(0,-a)$, Dir $y=a$
Length of Latus Rectum = $4a$
Q1
1. Find focus, axis, directrix, latus rectum: $y^2 = 12x$
Compare
Form $y^2 = 4ax$. Here $4a = 12 \Rightarrow a = 3$.
Focus
$(a, 0) \Rightarrow (3, 0)$
Axis
x-axis ($y=0$)
Directrix
$x = -a \Rightarrow x = -3$
L.R.
$4a = 12$
Focus(3,0), Axis: x, Dir: x=-3, LR: 12
Q2
2. Find parameters: $x^2 = 6y$
Compare
Form $x^2 = 4ay$. Here $4a = 6 \Rightarrow a = 6/4 = 3/2$.
Focus
$(0, a) \Rightarrow (0, 3/2)$
Axis
y-axis ($x=0$)
Directrix
$y = -a \Rightarrow y = -3/2$
L.R.
$4a = 6$
Focus(0, 3/2), Axis: y, Dir: y=-3/2, LR: 6
Q3
3. Find parameters: $y^2 = -8x$
Compare
Form $y^2 = -4ax$. Here $4a = 8 \Rightarrow a = 2$.
Focus
$(-a, 0) \Rightarrow (-2, 0)$
Axis
x-axis
Directrix
$x = a \Rightarrow x = 2$
L.R.
$4a = 8$
Focus(-2,0), Axis: x, Dir: x=2, LR: 8
Q4
4. Find parameters: $x^2 = -16y$
Compare
Form $x^2 = -4ay$. Here $4a = 16 \Rightarrow a = 4$.
Focus
$(0, -a) \Rightarrow (0, -4)$
Axis
y-axis
Directrix
$y = a \Rightarrow y = 4$
L.R.
$4a = 16$
Focus(0,-4), Axis: y, Dir: y=4, LR: 16
Q5
5. Find parameters: $y^2 = 10x$
Compare
$y^2 = 4ax \Rightarrow 4a = 10 \Rightarrow a = 2.5$.
Focus
$(2.5, 0)$
Directrix
$x = -2.5$
L.R.
10
Focus(2.5, 0), Dir: x=-2.5, LR: 10
Q6
6. Find parameters: $x^2 = -9y$
Compare
$x^2 = -4ay \Rightarrow 4a = 9 \Rightarrow a = 9/4$.
Focus
$(0, -9/4)$
Directrix
$y = 9/4$
L.R.
9
Focus(0, -2.25), Dir: y=2.25, LR: 9
Q7
7. Find equation: Focus (6, 0); Directrix x = -6.
Type
Focus is on positive x-axis. Directrix is to the left. $\Rightarrow y^2 = 4ax$.
Value a
Focus is $(a, 0)$, so $a = 6$.
Equation
$y^2 = 4(6)x = 24x$.
$y^2 = 24x$
Q8
8. Find equation: Focus (0, -3); Directrix y = 3.
Type
Focus on negative y-axis. $\Rightarrow x^2 = -4ay$.
Value a
Focus $(0, -a)$ is $(0, -3) \Rightarrow a = 3$.
Equation
$x^2 = -4(3)y$.
$x^2 = -12y$
Q9
9. Find equation: Vertex (0, 0); Focus (3, 0).
Type
Focus on positive x-axis. $\Rightarrow y^2 = 4ax$.
Value a
$a = 3$.
$y^2 = 12x$
Q10
10. Find equation: Vertex (0, 0); Focus (-2, 0).
Type
Focus on negative x-axis. $\Rightarrow y^2 = -4ax$.
Value a
$a = 2$ (Distance is positive).
$y^2 = -8x$
Q11
11. Vertex (0,0), passing through (2,3), axis along x-axis.
Type
Axis along x-axis means $y^2 = 4ax$ or $y^2 = -4ax$. Point (2,3) is in 1st quadrant, so $y^2 = 4ax$.
Substitute
Put $(2, 3)$ in $y^2 = 4ax$:
$(3)^2 = 4a(2) \Rightarrow 9 = 8a \Rightarrow a = 9/8$.
$(3)^2 = 4a(2) \Rightarrow 9 = 8a \Rightarrow a = 9/8$.
Equation
$y^2 = 4(9/8)x = \frac{9}{2}x$.
$2y^2 = 9x$
Q12
12. Vertex (0,0), passing through (5,2), symmetric w.r.t y-axis.
Type
Symmetric w.r.t y-axis means $x^2 = 4ay$ or $x^2 = -4ay$. Point (5,2) is in 1st quadrant (y>0), so $x^2 = 4ay$.
Substitute
Put $(5, 2)$ in $x^2 = 4ay$:
$(5)^2 = 4a(2) \Rightarrow 25 = 8a \Rightarrow a = 25/8$.
$(5)^2 = 4a(2) \Rightarrow 25 = 8a \Rightarrow a = 25/8$.
Equation
$x^2 = 4(\frac{25}{8})y = \frac{25}{2}y$.
$2x^2 = 25y$