Questions
4 questions
Difficulty
Medium-Hard
Importance
Core — never skip
Overview
Conic sections are the curves obtained by the intersection of a plane with a double right circular cone, forming the foundation of coordinate geometry. This topic is critical for board exams and competitive entrances due to its high density of formulas and logical integration across multiple shapes. Mastering the standard forms and their properties is essential for solving complex coordinate geometry problems.
Circle
A circle is the locus of a point moving such that its distance from a fixed point (center) is always constant. Standard equations are defined by the center-radius form, which is the most frequent starting point for conic section problems.
- Standard equation: (x-h)^2 + (y-k)^2 = r^2
- General form: x^2 + y^2 + 2gx + 2fy + c = 0
- Center: (-g, -f)
- Radius: sqrt(g^2 + f^2 - c)
- Diameter form: (x-x1)(x-x2) + (y-y1)(y-y2) = 0
Parabola
A parabola is the locus of a point equidistant from a fixed point (focus) and a fixed line (directrix). In exams, focus on identifying the orientation of the axis of symmetry and the corresponding vertex-focus distance.
- Right-opening: y^2 = 4ax
- Left-opening: y^2 = -4ax
- Upward: x^2 = 4ay
- Downward: x^2 = -4ay
- Latus Rectum length: 4a
- Directrix equation: x = -a (for y^2 = 4ax)
Ellipse
An ellipse represents the locus of points where the sum of distances from two fixed points (foci) is constant. Distinguish clearly between horizontal and vertical ellipses to avoid errors with the denominator values of a and b.
- Standard equation: x^2/a^2 + y^2/b^2 = 1 (a > b)
- Eccentricity: e = sqrt(1 - b^2/a^2)
- Foci: (+/- ae, 0)
- Length of Latus Rectum: 2b^2/a
- Sum of focal distances = 2a
Hyperbola
A hyperbola is the locus of points where the absolute difference of distances from two foci is constant. Pay special attention to the conjugate axis and the relationship between constants a, b, and c.
- Standard equation: x^2/a^2 - y^2/b^2 = 1
- Eccentricity: e = sqrt(1 + b^2/a^2)
- Foci: (+/- ae, 0)
- Relationship: c^2 = a^2 + b^2
- Asymptotes: y = +/- (b/a)x
Formula Sheet
Circle: (x-h)^2 + (y-k)^2 = r^2
Parabola: y^2 = 4ax
Ellipse: x^2/a^2 + y^2/b^2 = 1
Hyperbola: x^2/a^2 - y^2/b^2 = 1
Eccentricity Ellipse: e = sqrt(1 - b^2/a^2)
Eccentricity Hyperbola: e = sqrt(1 + b^2/a^2)
Latus Rectum Parabola: 4a
Latus Rectum Ellipse: 2b^2/a
Exam Tip
Always convert any conic equation into its standard form first; 90% of mistakes occur because students try to extract parameters 'a' and 'b' from unsimplified expressions.
Common Mistakes
- Confusing the standard equation of a horizontal ellipse (a > b) with a vertical ellipse, leading to wrong foci and directrix calculations.
- Forgetting to divide by the constant term on the right side to convert general equations into the standard form before identifying parameters like 'a' and 'b'.
- Mixing up the signs in the eccentricity formulas for ellipses (subtraction) versus hyperbolas (addition).
More Revision Notes
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