Revision Notes Chapter 11 Conic Sections Class 11 Maths

Essential Terminology

Conic Section

A curve obtained as the intersection of a plane with a double right circular cone.

Focus (F)

A fixed point in the plane used to define the geometric locus of a conic section.

Directrix

A fixed straight line in the plane, not passing through the focus, used in the locus definition of a conic section.

Eccentricity (e)

The constant ratio of the distance of a point on the conic from the focus to its perpendicular distance from the directrix (e = PF/PM).

Latus Rectum

A line segment perpendicular to the major axis/axis of symmetry, passing through the focus, with both endpoints on the conic curve.

Geometric Generation & Eccentricity Conditions

A conic section is the locus of a point $P$ which moves in a plane such that its distance from a fixed point (focus $F$) bears a constant ratio to its perpendicular distance from a fixed straight line (directrix $L$). This constant ratio is called eccentricity ($e$).

Based on the value of eccentricity, the conics are categorized as follows:

• Circle: $e = 0$ (obtained when the cutting plane is perpendicular to the axis of the cone).
• Parabola: $e = 1$ (the distance of any point on the curve to the focus is exactly equal to its distance to the directrix).
• Ellipse: $e < 1$ (the cutting plane is inclined to the axis but does not cross the base).
• Hyperbola: $e > 1$ (the plane cuts both nappes of the double cone).

Understanding these geometric definitions is essential to solving advanced shifting-of-origin problems and deriving standard forms.

Standard Equations and Properties Matrix

Key points / Must remember

• For a general circle with center (h, k) and radius r, the equation is: (x - h)^2 + (y - k)^2 = r^2.
• The general second-degree equation ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 represents a circle if a = b and h = 0.
• For a parabola y^2 = 4ax, the coordinates of the endpoints of the latus rectum are (a, 2a) and (a, -2a).
• For an ellipse with major axis along the x-axis (a > b), the relationship between semi-major axis (a), semi-minor axis (b), and eccentricity (e) is: b^2 = a^2(1 - e^2).
• For a hyperbola (x^2/a^2 - y^2/b^2 = 1), the relation is: b^2 = a^2(e^2 - 1). Here, e is always greater than 1.
• The focal distance of any point P(x, y) on the parabola y^2 = 4ax is (x + a).
• The sum of the focal distances of any point on an ellipse is constant and equal to the length of the major axis (2a).

Solved Revision Examples

• {"title":"Example 1: Finding Parabola parameters","description":"Find the coordinates of the focus, equation of the directrix, and the length of the latus rectum for the parabola $y^2 = -12x$.","steps":["Compare $y^2 = -12x$ with the standard form $y^2 = -4ax$. We get $4a = 12 \\implies a = 3$.","Since the parabola opens to the left, the focus is located at $(-a, 0) = (-3, 0)$.","The equation of the directrix is $x = a \\implies x = 3$.","The length of the latus rectum is $4a = 12$."]}
• {"title":"Example 2: Finding Ellipse parameters","description":"Find the eccentricity, coordinate of the foci, and latus rectum of the ellipse $\\frac{x^2}{25} + \\frac{y^2}{9} = 1$.","steps":["Identify $a^2 = 25 \\implies a = 5$ and $b^2 = 9 \\implies b = 3$. Here $a > b$, so the major axis lies along the x-axis.","Use the formula $b^2 = a^2(1 - e^2) \\implies 9 = 25(1 - e^2) \\implies 1 - e^2 = 9/25 \\implies e^2 = 16/25 \\implies e = 4/5$.","Foci are at $(\\pm ae, 0) = (\\pm 5 \\cdot (4/5), 0) = (\\pm 4, 0)$.","Length of Latus Rectum = $2b^2/a = 2(9)/5 = 18/5$ units."]}

Board Exam Trap & Scoring Tips

Beware of Shifted Ellipses and Hyperbolas:
Many students incorrectly apply the formula $b^2 = a^2(1 - e^2)$ to vertical ellipses without swapping $a$ and $b$. If the equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $b > a$, the major axis is along the y-axis, and the relation becomes $a^2 = b^2(1 - e^2)$. Always identify which denominator is larger before applying the formula! In the CBSE exam, sketching the basic rough coordinate diagram fetches you partial step marks even if your final numerical answer goes wrong.

Quick Revision Self-Check Qs

• Q1: What is the length of the latus rectum of the hyperbola $9x^2 - 16y^2 = 144$? A: Divide by 144 to get standard form: $x^2/16 - y^2/9 = 1$. Here $a^2 = 16 \implies a = 4$, and $b^2 = 9$. Length of latus rectum = $2b^2/a = 2(9)/4 = 4.5$.
• Q2: Find the equation of the circle whose center is at $(2, -3)$ and passing through the origin $(0,0)$. A: Radius $r = \sqrt{(2-0)^2 + (-3-0)^2} = \sqrt{4+9} = \sqrt{13}$. Equation: $(x-2)^2 + (y+3)^2 = 13 \implies x^2 + y^2 - 4x + 6y = 0$.
• Q3: If the directrix of a parabola is $y = 4$ and focus is $(0, -4)$, what is its standard equation? A: The focus lies on the negative y-axis, and the directrix is parallel to the x-axis. Thus, it opens downwards: $x^2 = -4ay$. Since $a = 4$, the equation is $x^2 = -16y$.
• Q4: Define the constant distance sum property of an Ellipse. A: For any point $P$ on an ellipse with foci $F_1$ and $F_2$, the sum of distances $PF_1 + PF_2$ is always constant and equal to $2a$ (length of the major axis).

Frequently Asked Questions