Conic Sections

Q1. (4 marks) A conic section is the locus of a point that moves such that the ratio of its distance from a fixed point (the focus) to its distance from a fixed line (the directrix) is a constant $e$ (the eccentricity). (2 marks: definition, focus, directrix, e)

• Parabola: $e = 1$
• Ellipse: $0 < e < 1$
• Hyperbola: $e > 1$ (2 marks, ~0.5 each; full 2 if all three correct)

Why: The eccentricity value fixes how "open" the curve is; $e=1$ balances the two distances exactly (parabola).

Q2. (4 marks) Compare $y^2 = 12x$ with $y^2 = 4ax$: $4a = 12 \Rightarrow a = 3$. (1)

• Focus: $(a, 0) = (3, 0)$ (1)
• Directrix: $x = -a \Rightarrow x = -3$ (1)
• Latus rectum length $= 4a = 12$ (1)

Why: Standard right-opening parabola parameters follow directly from $4a$.

Q3. (4 marks)

• Downward-opening parabola: $x^2 = -4ay$ (1.5)
• Focus: $(0, -a)$ (1)
• Reflective property: rays parallel to the axis reflect off the parabola through the focus (and vice versa). (1)
• Application: parabolic reflectors in telescopes / satellite dishes / car headlights. (0.5)

Why: The geometric reflection law combined with the parabola's shape concentrates incoming parallel rays at the focus.

Q4. (5 marks) Here $a^2 = 25 \Rightarrow a = 5$, $b^2 = 9 \Rightarrow b = 3$ (with $a>b$, major axis along $x$). (a) Semi-major axis $= 5$, semi-minor axis $= 3$ (1) (b) $c^2 = a^2 - b^2 = 25 - 9 = 16 \Rightarrow c = 4$; foci $(\pm 4, 0)$ (2) (c) $e = c/a = 4/5 = 0.8$ (1) (d) Latus rectum $= \dfrac{2b^2}{a} = \dfrac{2(9)}{5} = \dfrac{18}{5} = 3.6$ (1)

Why: Relation $c^2=a^2-b^2$ locates the foci; LR formula measures the chord through a focus perpendicular to the major axis.

Q5. (4 marks) Sum of focal radii property: for any point on an ellipse, the sum of distances to the two foci equals $2a$. (1) Point $(0,3)$, foci $(\pm 4, 0)$: $r_1 = \sqrt{(0-4)^2 + 3^2} = \sqrt{16+9} = \sqrt{25} = 5$ (1) $r_2 = \sqrt{(0+4)^2 + 3^2} = 5$ (1) Sum $= 5 + 5 = 10 = 2a = 2(5)$. Verified. (1)

Why: This constant-sum is the defining metric property of the ellipse.

Q6. (5 marks) Compare with $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$: $a^2 = 16, b^2 = 9$, so $a=4, b=3$.

• $c^2 = a^2 + b^2 = 16 + 9 = 25 \Rightarrow c = 5$; foci $(\pm 5, 0)$ (2)
• $e = c/a = 5/4 = 1.25$ (1.5)
• Asymptotes: $y = \pm \dfrac{b}{a}x = \pm \dfrac{3}{4}x$ (1.5)

Why: For a hyperbola $c^2 = a^2 + b^2$ (note the $+$), and asymptotes have slopes $\pm b/a$.

Q7. (4 marks) $xy = 16 \Rightarrow c^2 = 16 \Rightarrow c = 4$. Vertices lie on line $y = x$: $x^2 = 16 \Rightarrow x = \pm 4$, giving $(4, 4)$ and $(-4, -4)$. (2) Eccentricity of any rectangular hyperbola $= \sqrt{2}$. (2)

Why: A rectangular (equilateral) hyperbola has $a = b$, so $e = \sqrt{1 + b^2/a^2} = \sqrt{2}$; vertices are the nearest points to origin along $y=x$.

Q8. (4 marks) As $e \to 0$, the two foci of an ellipse merge at the centre and $b \to a$, so the ellipse becomes a circle; thus the circle is a degenerate conic with $e = 0$. (2) Parametric form: $x = a\cos\theta,\; y = a\sin\theta,\; \theta \in [0, 2\pi)$. (2)

Why: Zero eccentricity means both semi-axes equal, i.e. perfect circular symmetry.

Q9. (3 marks) $A = 3,\ B = 5,\ C = 2$. Discriminant $B^2 - 4AC = 25 - 24 = 1 > 0$. (2) Since $> 0$, the conic is a hyperbola (non-degenerate). (1)

Why: $B^2-4AC>0 \Rightarrow$ hyperbola, $=0 \Rightarrow$ parabola, $<0 \Rightarrow$ ellipse/circle.

Q10. (3 marks) (a) Parabola $y^2 = 4ax$: $x = at^2,\; y = 2at$. (1.5) (b) Ellipse: $x = a\cos\theta,\; y = b\sin\theta$. (1.5)

Why: Substituting these parameters back reproduces the Cartesian equations identically.

[
  {"claim":"Q2: latus rectum of y^2=12x is 12 and focus x-coord is 3","code":"a=Rational(12,4); LR=4*a; result=(a==3 and LR==12)"},
  {"claim":"Q4: ellipse x^2/25+y^2/9=1 has c=4, e=4/5, LR=18/5","code":"a2=25; b2=9; c=sqrt(a2-b2); e=c/sqrt(a2); LR=Rational(2*b2,5); result=(c==4 and e==Rational(4,5) and LR==Rational(18,5))"},
  {"claim":"Q5: focal radii sum at (0,3) equals 2a=10","code":"r1=sqrt((0-4)**2+3**2); r2=sqrt((0+4)**2+3**2); result=(r1+r2==10)"},
  {"claim":"Q6: hyperbola x^2/16-y^2/9=1 has c=5, e=5/4","code":"a2=16; b2=9; c=sqrt(a2+b2); e=c/sqrt(a2); result=(c==5 and e==Rational(5,4))"},
  {"claim":"Q9: discriminant of 3x^2+5xy+2y^2 is positive (hyperbola)","code":"A=3; B=5; C=2; disc=B**2-4*A*C; result=(disc==1 and disc>0)"}
]