Conic Sections: Parabola, Ellipse & Hyperbola

Focus-Directrix Definition

A conic section is the locus of a point $P$ moving such that its distance from a fixed point $S$ (Focus) bears a constant ratio $e$ (Eccentricity) to its perpendicular distance from a fixed line $L$ (Directrix): $$\frac{SP}{PM} = e$$

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📐 Interactive Conic Section Eccentricity Sandbox

Interactive Parametric Visualizer

Conic Section Eccentricity & Focus-Directrix Sandbox

Slide the eccentricity $e$ to witness the continuous geometric phase transition: Circle ($e=0$) $\to$ Ellipse ($0 < e < 1$) $\to$ Parabola ($e=1$) $\to$ Hyperbola ($e > 1$).

Eccentricity ($e$): 0.65 (Ellipse)

Semi-Latus Rectum Parameter ($p$): 60

Conic ClassificationEllipse

Focal Distance SP / PM Ratio0.65

Standard Cartesian Formx²/a² + y²/b² = 1 (b² = a²(1-e²))

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🚦 Step 0 Decision Tree: Conic Tangent & Normal Equations

🚦 Step 0 Decision TreeDeciding Parametric (t/θ) vs Slope (m) vs Point (T=0) Form

Click on your mathematical problem pattern below to instantly reveal the optimal, lowest-algebra solution path:

📐Optimal Solution Blueprint for: Definite Integral with Trigonometric Fraction

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Step 1: Check King’s Symmetry:Apply King’s property: $I = \int_a^b f(a+b-x) dx$. Write $2I = \int_a^b [f(x) + f(a+b-x)] dx$.

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Step 2: Add Integrals:Summing the integrands usually cancels the denominator completely, leaving $\int_a^b 1 dx = b - a$.

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Step 3: Solve for I:$2I = b - a \implies I = \frac{b - a}{2}$. Done in 3 lines without finding an antiderivative!

⚡ Pro-Tip / Exam Shortcut: For $\int_0^\pi x f(\sin x) dx$, King’s rule instantly eliminates the linear $x$ term, converting it to $\frac{\pi}{2} \int_0^\pi f(\sin x) dx$.

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1. The Standard Parabola ($y^2=4ax$)

• Focus: $(a,0)$
• Directrix: $x=-a$
• Latus Rectum Length: $4a$ (ends at $(a,2a),(a,-2a)$)
• Parametric Coordinates: $x=a{t}^2,y=2at$

Tangent and Normal in Parametric Form

• Tangent at $t$: $ty=x+a{t}^2$ (Slope $m=1/t$)
• Tangent in Slope Form ($m$): $y=mx+\frac{a}{m}$
• Normal at $t$: $y+tx=2at+a{t}^3$ (Slope $m=-t$)
• Normal in Slope Form ($m$): $y=mx-2am-a{m}^3$

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2. 4-Tier Scaffolded Hint Drill

💡 4-Tier Scaffolded Hint SystemJEE Advanced Problem: Common Tangent to Parabola and Ellipse

Try solving with Hint 1 before unlocking deeper tiers!

🟡 Tier 1: Core Mathematical Principle & Symmetry

Write the equation of tangent to the parabola $y^2 = 4ax$ in slope form: $y = m x + \frac{a}{m}$.

🟠 Tier 2: The Opening Move & Canonical Substitution

Recall the condition of tangency for the ellipse $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$: $c^2 = A^2 m^2 + B^2$.

🔵 Tier 3: Intermediate Milestone Equation

Equate $c = \frac{a}{m}$ into the ellipse tangency condition: $\left(\frac{a}{m}\right)^2 = A^2 m^2 + B^2$, yielding a biquadratic in $m$.

🟢 Tier 4: Complete Rigorous Solution & Answer

Solve for $m^2$ to obtain the common tangent slopes. Reject negative $m^2$ values for real tangents.

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3. Conic Section Formula & Danger Zone Cheat Sheet

📑 Quick Revision Matrix & Danger ZonesConic Section Tangent, Normal & Discriminant Conditions

Every governing formula with explicit mathematical conditions and failure danger zones:

Concept / Identity	Formula / Equation	Prerequisites	Where this Formula FAILS / Danger Zone
King's Property	$\int_a^b f(x) dx = \int_a^b f(a + b - x) dx$	$x \in [a, b]$	Fails if $f$ has non-integrable singularities within $(a, b)$. Useful primarily when $f(x) + f(a+b-x)$ cancels denominators.
Leibniz Integral Derivative	$\frac{d}{dx}\int_{u(x)}^{v(x)} f(x,t)dt = f(x,v)v' - f(x,u)u' + \int_u^v \frac{\partial f}{\partial x}dt$	$u, v \in C^1, \frac{\partial f}{\partial x} \in C^0$	Fails if limits $u(x), v(x)$ are discontinuous or if $\partial f/\partial x$ is not uniformly continuous on the domain.
Logarithmic Power Expansion	$\ln(x^2) = 2 \ln|x|$	$x \in \mathbb{R} \setminus \{0\}$	Writing $2\ln(x)$ without the modulus fails for all negative $x < 0$, stripping half the real domain!
Radical Product Identity	$\sqrt{a} \cdot \sqrt{b} = \sqrt{a b}$	$a \ge 0 \text{ or } b \ge 0$	Fails in $\mathbb{C}$ when both $a < 0$ and $b < 0$: $\sqrt{-1}\sqrt{-1} = i^2 = -1 \neq \sqrt{(-1)(-1)} = +1$.
AM-GM Inequality	$\frac{\sum_{i=1}^n a_i}{n} \ge \sqrt[n]{\prod_{i=1}^n a_i}$	$a_i > 0 \; (\forall i)$	Fails completely if any variable $a_i$ is negative or if equality is assumed without checking $a_1 = a_2 = \dots = a_n$.
Inverse Tangent Addition	$\arctan x + \arctan y = \arctan\left(\frac{x+y}{1-xy}\right)$	$x y < 1$	If $x > 0, y > 0$ and $x y > 1$, formula requires $+\pi$. If $x < 0, y < 0$ and $x y > 1$, formula requires $-\pi$.