Ellipse — Complete JEE Revision Capsule
Dense, exam-focused ellipse notes: standard forms, parametric machinery, tangents & normals, focal & directrix properties, director circle, chord formulas, tricks, common pitfalls and practice problems — all MathJax-ready.
$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$ (major axis along x if $a>b$)
$c^2=a^2-b^2,\ e=\\dfrac{c}{a}=\\sqrt{1-\\dfrac{b^2}{a^2}}$
Area $=\\pi ab$, Latus rectum $=\\dfrac{2b^2}{a}$
1. Standard Forms & Basic Definitions
Canonical ellipse (center at origin) with major axis along x: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\quad (a>b>0) \]
Major axis length = $2a$, minor axis length = $2b$, semi-major = $a$, semi-minor = $b$.
Foci: $(\pm c,0)$ where $c^2=a^2-b^2$.
Eccentricity: $e=\dfrac{c}{a}=\sqrt{1-\dfrac{b^2}{a^2}}$ (for $a>b$).
Directrices: $x=\pm \dfrac{a}{e}$ (lines perpendicular to major axis).
Vertical major axis: swap $x\leftrightarrow y$: \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \quad (a>b) \]
2. Parametric Form & Derivatives
Parametric coordinates (excellent for tangents, normals, lengths):
$$ x = a\cos t,\qquad y = b\sin t,\qquad t\in[0,2\pi). $$
Derivative (slope) at $(x_1,y_1)$:
$$ \frac{dy}{dx} = -\frac{b^2 x_1}{a^2 y_1} \quad\text{(provided } y_1\neq0\text{)}. $$
Using param: $$\frac{dy}{dx} = -\frac{b}{a}\cot t.$$
3. Tangent Equations (Common Forms)
Point form (at $(x_1,y_1)$ on ellipse):
$$ \boxed{\,\dfrac{x x_1}{a^2} + \dfrac{y y_1}{b^2} = 1 \,} $$
Parametric form (at parameter $t$):
$$ \boxed{\,\dfrac{x\cos t}{a} + \dfrac{y\sin t}{b} = 1\,} $$
Slope form (line $y=mx+c$ tangent iff):
$$ \boxed{\,c^2 = a^2 m^2 + b^2\,} $$ So tangents with slope $m$ are $y=mx \pm \sqrt{a^2 m^2 + b^2}$.
Condition: If $y=mx+c$ intersects ellipse, discriminant $=0$ for tangency; use above relation.
4. Normal Equations
Normal at parameter $t$ (useful in 3-normal problems):
$$ \boxed{\, a\sin t \, x - b\cos t \, y + (a^2-b^2)\sin t\cos t = 0 \,} $$
(One may also derive normal using slope $m_n = -\dfrac{a^2}{b^2}\dfrac{x_1}{y_1}$ and point-slope.)
5. Latus Rectum, Area, Perimeter (Formulas)
- Latus rectum (through focus): length $= \dfrac{2b^2}{a}$.
- Area: $A = \pi a b$.
- Approx perimeter (Ramanujan): $$ P \approx \pi\left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]. $$
6. Focal Properties & Directrix
Definition in terms of focus & directrix (eccentricity $e$):
Distance to focus = $e\times$ distance to corresponding directrix.
Directrices (for major axis along x): \[ x = \pm \frac{a}{e}. \]
Reflection property: Ray from one focus reflects to the other (sum of distances to foci is constant = $2a$).
7. Director Circle & Orthogonality
Director circle: locus of points from which pair of tangents to ellipse are orthogonal:
$$ \boxed{\, x^2 + y^2 = a^2 + b^2 \,} $$
Orthogonal circles: Two conics orthogonal if their tangents at intersection are perpendicular; for circles use condition, for conics use gradients.
8. Chords, Chord of Contact & Midpoint Locus
Chord joining parameters $t_1,t_2$ (param method): midpoint and chord relations can be derived; product/sum of parameters used for focal chords.
Chord of contact from point $(x_1,y_1)$ (pair of tangents from an external point):
$$ \boxed{\,\dfrac{x x_1}{a^2} + \dfrac{y y_1}{b^2} = 1 \,} \quad\text{(same as tangent template, with }(x_1,y_1)\text{ on polar).} $$
Midpoint locus: Many problems ask locus of midpoints of parallel chords → result is a line parallel to corresponding axis. For example, midpoints of chords parallel to y-axis have x = constant.
9. Condition for Point & Regions
For point $P(x_1,y_1)$ compute:
$$ S = \dfrac{x_1^2}{a^2} + \dfrac{y_1^2}{b^2} - 1. $$
If $S<0 ellipse="" inside="" on="" point="">0$ outside ellipse.
10. Useful Identities & Quick Tricks
- Sum of distances to foci from any point on ellipse $=2a$ (constant).
- Tangent at $(x_1,y_1)$ can be written quickly using point form: substitute in formula.
- To get tangent with given slope $m$, use $c^2=a^2m^2+b^2$ → find $c$.
- For intersection with line $y=mx+c$, solve quadratic; D<0 none, D=0 tangent, D>0 secant.
- When converting rotated ellipses (Bxy term present), use rotation of axes: $\tan 2\theta = \dfrac{B}{A-C}$.
11. Worked Examples (High-yield)
Example 1: For ellipse $\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$, find foci, eccentricity, latus rectum.
Solution: $a^2=9\\Rightarrow a=3$, $b^2=4\\Rightarrow b=2$. $c^2=a^2-b^2=9-4=5\\Rightarrow c=\\sqrt5$. $e=c/a=\\dfrac{\\sqrt5}{3}$. Latus rectum $=\\dfrac{2b^2}{a}=\\dfrac{2\\cdot4}{3}=\\dfrac{8}{3}$.
Example 2: Find tangent at point corresponding to $t=\\pi/6$ on $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$.
Solution: Parametric point: $x=a\\cos t=a(\\sqrt3/2)$, $y=b\\sin t=b/2$. Use point form: $$\\dfrac{x x_1}{a^2}+\\dfrac{y y_1}{b^2}=1.$$ Simplify to obtain explicit tangent.
Example 3: Tangent with slope $m=1$ to ellipse $\\dfrac{x^2}{16}+\\dfrac{y^2}{9}=1$ → find equation(s).
Solution: Use $c^2 = a^2 m^2 + b^2 = 16\\cdot1 + 9 =25$. So $c=\\pm5$. Tangents: $y=x\\pm5$.
12. Common JEE Traps & How to Avoid
- Trap: Confusing a & b (which is larger) — always set $a$ as semi-major and check $a>b$ or rotate axes accordingly.
- Trap: Wrong directrix sign/position — use $x=\\pm a/e$ for horizontal major axis.
- Trap: Using circle formulas — ellipses have sum of distances constant, not product.
- Tip: For chord-midpoint loci use parametric midpoint formula; many past JEE problems follow this route.
13. Practice Problems (Try first)
- Find equation of tangent to $\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$ at point where $t=\\pi/4$.
- For ellipse $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$, show that product of slopes of tangents at ends of any focal chord = \\(-\\dfrac{b^2}{a^2}\\).
- Find director circle of ellipse $\\dfrac{x^2}{16}+\\dfrac{y^2}{4}=1$.
- Find equation(s) of tangent(s) to $\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$ parallel to line $y=2x+3$.
- Find the length of latus rectum for ellipse with equation $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$.
Show Answers & Hints
- Parametric point: $x=a\\cos t, y=b\\sin t$; substitute into point form tangent.
- Use param for focal chord ($t_1t_2=-1$) and slopes $m_i=-\\dfrac{b^2}{a^2}\\dfrac{x_i}{y_i}$ or use algebraic elimination.
- Director circle: $x^2+y^2=a^2+b^2$ → here $=16+4=20$.
- Tangents parallel to slope 2: use $c^2=a^2 m^2 + b^2$ with $m=2$ to find $c$ and write $y=2x\\pm c$.
- Latus rectum $=\\dfrac{2b^2}{a}$ (derive from focal chord properties).
14. One-line Takeaways
- Always check which axis is major — set $a$ as semi-major by convention.
- Parametric form $x=a\\cos t, y=b\\sin t$ is your fastest tool for tangents/normals/lengths.
- Focal property: sum of distances to foci = $2a$; directrices at $x=\\pm a/e$.
- Director circle: $x^2+y^2=a^2+b^2$ — quick test for orthogonal tangents.
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