Level 1 — RecognitionConic Sections

Conic Sections

30 marksprintable — key stays hidden on paper

Time: 20 minutes Total Marks: 30

Section A — Multiple Choice (1 mark each)

Choose the correct option.

Q1. The eccentricity of a circle is: (a) $e=1$ (b) $0<e<1$ (c) $e=0$ (d) $e>1$

Q2. For the parabola $y^2 = 12x$ , the focus is at: (a) $(3,0)$ (b) $(0,3)$ (c) $(-3,0)$ (d) $(6,0)$

Q3. The length of the latus rectum of $y^2 = 4ax$ is: (a) $a$ (b) $2a$ (c) $4a$ (d) $a/2$

Q4. For the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{9}=1$ , the value of $a$ (semi-major axis) is: (a) $3$ (b) $4$ (c) $5$ (d) $9$

Q5. For an ellipse, the sum of focal radii of any point equals: (a) $a$ (b) $2a$ (c) $2b$ (d) $b^2/a$

Q6. A conic with $e>1$ is a: (a) circle (b) parabola (c) ellipse (d) hyperbola

Q7. The asymptotes of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ are: (a) $y=\pm \frac{a}{b}x$ (b) $y=\pm \frac{b}{a}x$ (c) $y=\pm x$ (d) $x=\pm a$

Q8. The rectangular hyperbola has equation: (a) $x^2+y^2=c^2$ (b) $xy=c^2$ (c) $x^2-y^2=1$ (d) $y^2=4ax$

Q9. The directrix of the parabola $y^2=4ax$ is: (a) $x=a$ (b) $x=-a$ (c) $y=a$ (d) $y=-a$

Q10. Parametric coordinates of a point on the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ are: (a) $(a\sec\theta, b\tan\theta)$ (b) $(a\cos\theta, b\sin\theta)$ (c) $(at^2, 2at)$ (d) $(a\cos\theta, a\sin\theta)$

Q11. For the general second-degree equation $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ , the conic is a parabola when: (a) $B^2-4AC<0$ (b) $B^2-4AC=0$ (c) $B^2-4AC>0$ (d) $B^2-4AC=1$

Q12. The eccentricity of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$ is: (a) $\frac{3}{5}$ (b) $\frac{4}{5}$ (c) $\frac{5}{3}$ (d) $\frac{1}{5}$

Section B — Matching (5 marks)

Q13. Match each conic/property in Column I with Column II. (1 mark each)

Column I	Column II
(i) $x^2=4ay$	(P) hyperbola
(ii) $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$	(Q) circle
(iii) $e=0$	(R) parabola opening upward
(iv) reflective property used in telescopes	(S) sum of focal radii $=2a$
(v) ellipse	(T) parabola

Section C — True / False with Justification (2 marks each)

State True or False. 1 mark for the verdict, 1 mark for a correct one-line justification.

Q14. For the parabola, all rays parallel to the axis reflect through the focus.

Q15. In a hyperbola, the difference of the focal radii of any point is constant and equals $2b$ .

Q16. Kepler showed that planetary orbits are ellipses with the Sun at one focus.

Q17. For the ellipse $\dfrac{x^2}{16}+\dfrac{y^2}{25}=1$ , the major axis lies along the $x$ -axis.

Q18. The eccentricity of a hyperbola is related to $a,b$ by $b^2=a^2(e^2-1)$ .

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1. (c) $e=0$ . A circle is the degenerate conic with zero eccentricity. [1]

Q2. (a) $(3,0)$ . Compare $y^2=12x$ with $y^2=4ax\Rightarrow 4a=12\Rightarrow a=3$ ; focus $(a,0)=(3,0)$ . [1]

Q3. (c) $4a$ . Standard result for $y^2=4ax$ . [1]

Q4. (c) $5$ . $a^2=25\Rightarrow a=5$ (larger denominator gives semi-major axis). [1]

Q5. (b) $2a$ . Defining focal-sum property of ellipse. [1]

Q6. (d) hyperbola. $e>1$ characterises a hyperbola. [1]

Q7. (b) $y=\pm\frac{b}{a}x$ . Asymptotes of the standard horizontal hyperbola. [1]

Q8. (b) $xy=c^2$ . Rectangular (equilateral) hyperbola. [1]

Q9. (b) $x=-a$ . Directrix of $y^2=4ax$ . [1]

Q10. (b) $(a\cos\theta, b\sin\theta)$ . Standard ellipse parametrisation. [1]

Q11. (b) $B^2-4AC=0$ . Discriminant zero ⇒ parabola. [1]

Q12. (a) $\frac{3}{5}$ . $b^2=a^2(1-e^2)\Rightarrow 16=25(1-e^2)\Rightarrow e^2=9/25\Rightarrow e=3/5$ . [1]

Section B

Q13. (i)→(R), (ii)→(P), (iii)→(Q), (iv)→(T), (v)→(S). [5 × 1] Why: $x^2=4ay$ opens upward; $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is a hyperbola; $e=0$ ⇒ circle; the parabola's reflective property is used in telescopes; the ellipse is defined by focal-radii sum $=2a$ .

Section C (verdict 1 + justification 1)

Q14. True. By the parabola's reflective property, incoming rays parallel to the axis all pass through the focus after reflection. [2]

Q15. False. The constant difference of focal radii equals $2a$ (transverse axis length), not $2b$ . [2]

Q16. True. Kepler's first law: each planet moves in an ellipse with the Sun at one focus. [2]

Q17. False. Here $b^2=25>a^2=16$ , so the larger axis is along the $y$ -axis; major axis is vertical. [2]

Q18. True. For a hyperbola $b^2=a^2(e^2-1)$ , equivalently $e^2=1+b^2/a^2>1$ . [2]

[
  {"claim":"Q2 parabola y^2=12x has focus (3,0)","code":"a=Rational(12,4); result=(a==3)"},
  {"claim":"Q12 ellipse x^2/25+y^2/16 has e=3/5","code":"e2=1-Rational(16,25); result=(sqrt(e2)==Rational(3,5))"},
  {"claim":"Q3 latus rectum of y^2=4ax equals 4a","code":"a=symbols('a',positive=True); result=(4*a==4*a)"},
  {"claim":"Q18 hyperbola relation b^2=a^2(e^2-1) gives e>1","code":"a,b=symbols('a b',positive=True); e=sqrt(1+b**2/a**2); result=(simplify(b**2-a**2*(e**2-1))==0)"}
]