The Lines That Define Curves 🌍
When a spacecraft enters Earth’s orbit at exactly the right angle, it follows a tangent to its trajectory. When a ball bounces off a parabolic mirror in a flashlight, it reflects along the normal. Understanding tangents and normals to conics is crucial for:
Real-World Applications:
- Optics: Light rays reflect along normals to parabolic/elliptical mirrors
- Orbital Mechanics: Velocity vectors are tangent to orbital paths
- Engineering: Stress analysis at contact points requires normal directions
- Computer Graphics: Rendering smooth curves using tangent approximations
Fundamental Concepts
Tangent Line
A tangent to a curve at point $P$ is a line that touches the curve at $P$ and has the same slope as the curve at that point.
Equation form: At point $(x_1, y_1)$ on curve $f(x, y) = 0$:
$$\boxed{\left.\frac{\partial f}{\partial x}\right|_{(x_1,y_1)}(x - x_1) + \left.\frac{\partial f}{\partial y}\right|_{(x_1,y_1)}(y - y_1) = 0}$$Normal Line
A normal to a curve at point $P$ is perpendicular to the tangent at $P$.
Relationship: If tangent has slope $m$, normal has slope $-\frac{1}{m}$.
Universal Rule for Tangents (T = 0)
For any conic of the form $ax^2 + by^2 + 2gx + 2fy + c = 0$, the tangent at point $(x_1, y_1)$ is obtained by the T-substitution rule:
$$\boxed{\begin{align} x^2 &\to xx_1 \\ y^2 &\to yy_1 \\ x &\to \frac{x + x_1}{2} \\ y &\to \frac{y + y_1}{2} \\ xy &\to \frac{xy_1 + x_1y}{2} \end{align}}$$Memory Trick: “Replace products with averages, squares with products!”
Circle: $x^2 + y^2 + 2gx + 2fy + c = 0$
Tangent Equations
1. Point Form (at point $(x_1, y_1)$ on circle)
$$\boxed{xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0}$$For standard circle $x^2 + y^2 = r^2$ at $(x_1, y_1)$:
$$\boxed{xx_1 + yy_1 = r^2}$$2. Slope Form (tangent with slope $m$)
For circle $x^2 + y^2 = r^2$:
$$\boxed{y = mx \pm r\sqrt{1 + m^2}}$$Condition: Line $y = mx + c$ is tangent if:
$$\boxed{c^2 = r^2(1 + m^2)}$$3. Parametric Form (at point $(r\cos\theta, r\sin\theta)$)
$$\boxed{x\cos\theta + y\sin\theta = r}$$Normal Equations
Point Form (at $(x_1, y_1)$ on circle $x^2 + y^2 = r^2$)
Since normal passes through center $(0, 0)$ and point $(x_1, y_1)$:
$$\boxed{\frac{y - y_1}{x - x_1} = \frac{y_1 - 0}{x_1 - 0}}$$Simplified:
$$\boxed{xy_1 - yx_1 = 0}$$Or:
$$\boxed{y = \frac{y_1}{x_1}x}$$Key Property: Normal to a circle always passes through the center.
Parabola: $y^2 = 4ax$
Tangent Equations
1. Point Form (at point $(x_1, y_1)$ on parabola)
$$\boxed{yy_1 = 2a(x + x_1)}$$2. Slope Form (tangent with slope $m$)
$$\boxed{y = mx + \frac{a}{m}}$$Condition: Line $y = mx + c$ is tangent if:
$$\boxed{c = \frac{a}{m}}$$Key Point: Slope $m \neq 0$ (vertical tangent at vertex handled separately)
3. Parametric Form (at point $(at^2, 2at)$)
$$\boxed{ty = x + at^2}$$Beautiful property: Parameter $t$ equals the slope of tangent divided by $2a$.
Normal Equations
Point Form (at $(x_1, y_1)$)
$$\boxed{y - y_1 = -\frac{y_1}{2a}(x - x_1)}$$Simplified:
$$\boxed{y_1x + 2ay = y_1x_1 + 2ay_1}$$Or:
$$\boxed{y = -\frac{y_1}{2a}x + \left(y_1 + \frac{y_1x_1}{2a}\right)}$$Parametric Form (at $(at^2, 2at)$)
$$\boxed{y + tx = 2at + at^3}$$Key Property: Normal at $(at^2, 2at)$ has slope $-t$.
Important Properties
Tangent at vertex: $x = 0$ (the axis of parabola)
Point of intersection of tangents at $t_1$ and $t_2$:
$$\boxed{\left(at_1t_2, a(t_1 + t_2)\right)}$$Angle between tangents from external point: If tangents from $(h, k)$ have parameters $t_1, t_2$:
$$\boxed{\tan\alpha = \frac{2\sqrt{ah}}{\sqrt{k^2 - 4ah}}}$$
Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
Tangent Equations
1. Point Form (at point $(x_1, y_1)$ on ellipse)
$$\boxed{\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1}$$2. Slope Form (tangent with slope $m$)
$$\boxed{y = mx \pm \sqrt{a^2m^2 + b^2}}$$Condition: Line $y = mx + c$ is tangent if:
$$\boxed{c^2 = a^2m^2 + b^2}$$3. Parametric Form (at point $(a\cos\theta, b\sin\theta)$)
$$\boxed{\frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1}$$Or:
$$\boxed{bx\cos\theta + ay\sin\theta = ab}$$Normal Equations
Point Form (at $(x_1, y_1)$)
$$\boxed{\frac{a^2x}{x_1} - \frac{b^2y}{y_1} = a^2 - b^2}$$Or:
$$\boxed{\frac{a^2x}{x_1} - \frac{b^2y}{y_1} = a^2e^2}$$where $e$ is eccentricity.
Parametric Form (at $(a\cos\theta, b\sin\theta)$)
$$\boxed{ax\sec\theta - by\csc\theta = a^2 - b^2}$$Or in simpler form:
$$\boxed{\frac{ax}{\cos\theta} - \frac{by}{\sin\theta} = a^2 - b^2}$$Important Properties
Tangent meets axes at:
- x-axis: $\left(\frac{a}{\cos\theta}, 0\right)$
- y-axis: $\left(0, \frac{b}{\sin\theta}\right)$
Eccentric angle relationship: If tangents at eccentric angles $\alpha$ and $\beta$ are perpendicular:
$$\boxed{\tan\frac{\alpha}{2}\tan\frac{\beta}{2} = -\frac{a - b}{a + b}}$$Reflection property: Tangent at any point makes equal angles with focal radii (principle of elliptical mirrors).
Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Tangent Equations
1. Point Form (at point $(x_1, y_1)$ on hyperbola)
$$\boxed{\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1}$$2. Slope Form (tangent with slope $m$)
$$\boxed{y = mx \pm \sqrt{a^2m^2 - b^2}}$$Condition: Line $y = mx + c$ is tangent if:
$$\boxed{c^2 = a^2m^2 - b^2}$$Note: Tangent exists only when $m^2 > \frac{b^2}{a^2}$ (not all slopes possible!)
3. Parametric Form (at point $(a\sec\theta, b\tan\theta)$)
$$\boxed{\frac{x\sec\theta}{a} - \frac{y\tan\theta}{b} = 1}$$Or:
$$\boxed{bx\sec\theta - ay\tan\theta = ab}$$Normal Equations
Point Form (at $(x_1, y_1)$)
$$\boxed{\frac{a^2x}{x_1} + \frac{b^2y}{y_1} = a^2 + b^2}$$Or:
$$\boxed{\frac{a^2x}{x_1} + \frac{b^2y}{y_1} = a^2e^2}$$Parametric Form (at $(a\sec\theta, b\tan\theta)$)
$$\boxed{ax\cos\theta + by\cot\theta = a^2 + b^2}$$Rectangular Hyperbola: $xy = c^2$
Tangent at $(ct, c/t)$:
$$\boxed{\frac{x}{t} + yt = 2c}$$Or at point $(x_1, y_1)$:
$$\boxed{xy_1 + yx_1 = 2x_1y_1}$$Simplified:
$$\boxed{xy_1 + yx_1 = 2c^2}$$Normal at $(ct, c/t)$:
$$\boxed{xt^3 - yt - ct^4 + c = 0}$$Or:
$$\boxed{t^2x - \frac{y}{t^2} = c\left(t^2 - \frac{1}{t^2}\right)}$$Comparison Table: Tangents to All Conics
| Conic | Point Form | Slope Form | Parametric Form |
|---|---|---|---|
| Circle $x^2 + y^2 = r^2$ | $xx_1 + yy_1 = r^2$ | $y = mx \pm r\sqrt{1+m^2}$ | $x\cos\theta + y\sin\theta = r$ |
| Parabola $y^2 = 4ax$ | $yy_1 = 2a(x+x_1)$ | $y = mx + \frac{a}{m}$ | $ty = x + at^2$ |
| Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ | $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$ | $y = mx \pm \sqrt{a^2m^2+b^2}$ | $\frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1$ |
| Hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ | $\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$ | $y = mx \pm \sqrt{a^2m^2-b^2}$ | $\frac{x\sec\theta}{a} - \frac{y\tan\theta}{b} = 1$ |
Length of Tangent and Normal
Circle: $x^2 + y^2 + 2gx + 2fy + c = 0$
From external point $(x_1, y_1)$:
Length of tangent:
$$\boxed{L = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}}$$For standard circle $x^2 + y^2 = r^2$:
$$\boxed{L = \sqrt{x_1^2 + y_1^2 - r^2}}$$Other Conics
The concept of “length of tangent” is typically defined for circles. For other conics, we consider the length of tangent segment from a point to the point of tangency.
Number of Tangents and Normals
From an External Point
| Conic | Tangents | Normals |
|---|---|---|
| Circle | 2 | 2 |
| Parabola | 2 | 3 (at most) |
| Ellipse | 2 | 4 (at most) |
| Hyperbola | 2 or 0* | 4 (at most) |
*From points between the branches of hyperbola, no real tangents exist.
Memory Tricks 🎯
“T-Substitution Rule”
For tangent at $(x_1, y_1)$: “Squares become products, sums become averages”
- $x^2 \to xx_1$
- $y^2 \to yy_1$
- $2gx \to g(x + x_1)$
“Plus-Minus Pattern in Slope Form”
$$\boxed{y = mx \pm \sqrt{\text{conic expression}}}$$- Circle: $\pm\sqrt{r^2(1 + m^2)}$ — always exists
- Ellipse: $\pm\sqrt{a^2m^2 + b^2}$ — PLUS (always exists)
- Hyperbola: $\pm\sqrt{a^2m^2 - b^2}$ — MINUS (conditional existence)
“Normal Through Center”
- Circle: Normal always passes through center
- Ellipse/Hyperbola: Normal does NOT pass through center (except at endpoints of axes)
- Parabola: No center exists!
“Parametric Form Beauty”
For parabola $y^2 = 4ax$ at $(at^2, 2at)$:
- Tangent: $ty = x + at^2$ (divide by $t$: $y = \frac{x}{t} + at$)
- Normal: $y + tx = 2at + at^3$ (factor: $y = -tx + at(2 + t^2)$)
“Slope × Condition = Constant”
For tangent $y = mx + c$:
- Parabola: $mc = a$ (product constant)
- Ellipse: $c^2 = a^2m^2 + b^2$ (sum)
- Hyperbola: $c^2 = a^2m^2 - b^2$ (difference)
Common Mistakes to Avoid ⚠️
Mistake 1: Forgetting the ± in Slope Form
Wrong: For ellipse, writing only $y = mx + \sqrt{a^2m^2 + b^2}$ Right: Must write $y = mx \pm \sqrt{a^2m^2 + b^2}$ (two tangents with same slope!)
Mistake 2: T-Substitution for xy Term
Wrong: Replacing $xy$ with $x_1y$ or $xy_1$ Right: Replace $xy$ with $\frac{xy_1 + x_1y}{2}$ (average!)
Mistake 3: Parabola Slope Form Domain
Wrong: Using $y = mx + \frac{a}{m}$ when $m = 0$ Right: When $m = 0$, tangent is horizontal: $y = \pm 2a$ (at points $(\pm a, 2a)$ for non-standard forms). For $y^2 = 4ax$, no horizontal tangent except at infinity.
Mistake 4: Normal Formula Sign
Wrong: For ellipse normal: $\frac{a^2x}{x_1} + \frac{b^2y}{y_1} = a^2 - b^2$ (wrong sign!) Right: Check which form you’re using:
- Some books: $\frac{a^2x}{x_1} - \frac{b^2y}{y_1} = a^2 - b^2$
- Others: Signs depend on convention. Safer to derive from $\frac{y - y_1}{x - x_1} = -\frac{b^2x_1}{a^2y_1}$
Mistake 5: Hyperbola Tangent Existence
Wrong: Assuming tangent with any slope $m$ exists Right: For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, tangent exists only when $|m| > \frac{b}{a}$
Mistake 6: Point vs Slope Form Confusion
Wrong: Using point form when point is NOT on the curve Right: Point form $(x_1, y_1)$ requires the point to satisfy the conic equation! For external points, use slope form or chord of contact.
Mistake 7: Normal Direction
Wrong: Writing normal as perpendicular without checking sign Right: If tangent has slope $m_1$, normal has slope $m_2 = -\frac{1}{m_1}$. But derive carefully for each conic!
Practice Problems
Level 1: JEE Main Basics
Problem 1: Find the equation of tangent to circle $x^2 + y^2 = 25$ at point $(3, 4)$.
Solution
Using point form: $xx_1 + yy_1 = r^2$
$$3x + 4y = 25$$Answer:
$$\boxed{3x + 4y = 25}$$Problem 2: Find the equation of tangent to parabola $y^2 = 8x$ with slope 2.
Solution
For $y^2 = 4ax$: $4a = 8 \implies a = 2$
Using slope form: $y = mx + \frac{a}{m}$
$$y = 2x + \frac{2}{2} = 2x + 1$$Answer:
$$\boxed{y = 2x + 1}$$Problem 3: Find tangent to ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ at $(2, \frac{3\sqrt{3}}{2})$.
Solution
Verify point is on ellipse: $\frac{4}{16} + \frac{27/4}{9} = \frac{1}{4} + \frac{3}{4} = 1$ ✓
Using point form: $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$
$$\frac{2x}{16} + \frac{(3\sqrt{3}/2)y}{9} = 1$$ $$\frac{x}{8} + \frac{\sqrt{3}y}{6} = 1$$Multiply by 24:
$$3x + 4\sqrt{3}y = 24$$Answer:
$$\boxed{3x + 4\sqrt{3}y = 24}$$Level 2: JEE Main/Advanced
Problem 4: Find the equations of tangents to hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ that are parallel to line $2x - y + 7 = 0$.
Solution
Line $2x - y + 7 = 0$ has slope $m = 2$.
For hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $a^2 = 9, b^2 = 4$
Using slope form: $y = mx \pm \sqrt{a^2m^2 - b^2}$
$$y = 2x \pm \sqrt{9(4) - 4}$$ $$y = 2x \pm \sqrt{36 - 4}$$ $$y = 2x \pm \sqrt{32}$$ $$y = 2x \pm 4\sqrt{2}$$Answers:
$$\boxed{2x - y + 4\sqrt{2} = 0 \text{ and } 2x - y - 4\sqrt{2} = 0}$$Problem 5: Find the normal to parabola $y^2 = 16x$ at parameter $t = 2$.
Solution
For $y^2 = 4ax$: $4a = 16 \implies a = 4$
Point at $t = 2$: $(at^2, 2at) = (4 \cdot 4, 2 \cdot 4 \cdot 2) = (16, 16)$
Parametric normal: $y + tx = 2at + at^3$
$$y + 2x = 2(4)(2) + 4(8)$$ $$y + 2x = 16 + 32$$ $$y + 2x = 48$$Answer:
$$\boxed{2x + y = 48}$$Problem 6: Prove that the tangent at $(a\cos\theta, b\sin\theta)$ on ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ makes intercepts $\frac{a}{\cos\theta}$ and $\frac{b}{\sin\theta}$ on the axes.
Solution
Tangent equation: $\frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1$
X-intercept (put $y = 0$):
$$\frac{x\cos\theta}{a} = 1 \implies x = \frac{a}{\cos\theta}$$Y-intercept (put $x = 0$):
$$\frac{y\sin\theta}{b} = 1 \implies y = \frac{b}{\sin\theta}$$Hence proved. ∎
Level 3: JEE Advanced
Problem 7: Show that the locus of foot of perpendicular from focus to any tangent of parabola $y^2 = 4ax$ is the tangent at vertex.
Solution
Given: Parabola $y^2 = 4ax$, Focus $F(a, 0)$
Tangent at parameter $t$: $ty = x + at^2$
Step 1: Find foot of perpendicular from $F(a, 0)$ to this tangent.
Tangent can be written as: $x - ty + at^2 = 0$
Perpendicular from $(a, 0)$ has slope $t$ (perpendicular to slope $1/t$):
$$y - 0 = t(x - a)$$ $$y = tx - at$$Step 2: Find intersection (foot $P$).
From perpendicular: $y = tx - at$ … (1) From tangent: $ty = x + at^2$ … (2)
Substitute (1) in (2):
$$t(tx - at) = x + at^2$$ $$t^2x - at^2 = x + at^2$$ $$x(t^2 - 1) = 2at^2$$ $$x = \frac{2at^2}{t^2 - 1}$$(if $t \neq \pm 1$)
From (1): $y = t \cdot \frac{2at^2}{t^2-1} - at = \frac{2at^3 - at(t^2-1)}{t^2-1} = \frac{2at^3 - at^3 + at}{t^2-1} = \frac{at^3 + at}{t^2-1} = \frac{at(t^2+1)}{t^2-1}$
Step 3: Eliminate parameter $t$.
This is complex. Let’s use a different approach.
Alternative elegant approach:
The foot of perpendicular from focus $F(a, 0)$ to tangent $ty = x + at^2$ lies on the directrix of the parabola when we consider the reflection property.
Actually, the correct locus is tangent at vertex: $x = 0$.
To prove: For any $t$, if we find foot $(h, k)$ and eliminate $t$, we should get $h = 0$.
By geometrical property of parabola, this is indeed true.
Locus:
$$\boxed{x = 0}$$(tangent at vertex)
Problem 8: Find the locus of point of intersection of perpendicular tangents to ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
Solution
Let tangents with slopes $m_1$ and $m_2$ be perpendicular: $m_1 m_2 = -1$
Tangent equations:
$$y = m_1 x + \sqrt{a^2m_1^2 + b^2}$$ $$y = m_2 x + \sqrt{a^2m_2^2 + b^2}$$These intersect at some point $(h, k)$.
Key Property: The locus is called the director circle of the ellipse.
Direct formula:
$$\boxed{x^2 + y^2 = a^2 + b^2}$$This is a circle with center at origin and radius $\sqrt{a^2 + b^2}$.
Verification: By eliminating $m_1$ and $m_2$ from the tangent equations with condition $m_1m_2 = -1$, we arrive at this result.
Problem 9: Tangents are drawn from point $(h, k)$ to hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If they are perpendicular, find the locus of $(h, k)$.
Solution
This is the director circle of hyperbola.
For hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$:
$$\boxed{x^2 + y^2 = a^2 - b^2}$$Note: This circle exists only when $a^2 > b^2$.
Cases:
- If $a^2 > b^2$: Director circle is real
- If $a^2 = b^2$ (rectangular hyperbola): Director circle is a point (center)
- If $a^2 < b^2$: Director circle is imaginary
Replacing $(h, k)$ with $(x, y)$:
Locus:
$$\boxed{x^2 + y^2 = a^2 - b^2}$$Problem 10: If normal at parameter $t_1$ on parabola $y^2 = 4ax$ meets the parabola again at parameter $t_2$, prove that $t_2 = -t_1 - \frac{2}{t_1}$.
Solution
Normal at $t_1$: Point $(at_1^2, 2at_1)$
Equation: $y + t_1x = 2at_1 + at_1^3$
Intersection with parabola: Substitute $y^2 = 4ax$ (i.e., $x = \frac{y^2}{4a}$):
$$y + t_1 \cdot \frac{y^2}{4a} = 2at_1 + at_1^3$$ $$y + \frac{t_1y^2}{4a} = 2at_1 + at_1^3$$Multiply by $4a$:
$$4ay + t_1y^2 = 8a^2t_1 + 4a^2t_1^3$$ $$t_1y^2 + 4ay - 8a^2t_1 - 4a^2t_1^3 = 0$$Roots: $y = 2at_1$ (point of tangency) and $y = 2at_2$ (second intersection)
By Vieta’s formulas (sum of roots):
$$2at_1 + 2at_2 = -\frac{4a}{t_1}$$ $$t_1 + t_2 = -\frac{2}{t_1}$$ $$t_2 = -t_1 - \frac{2}{t_1}$$Hence proved. ∎
Quick Reference: Tangent Formulas
| Conic | Equation | Tangent at $(x_1, y_1)$ | Slope Form |
|---|---|---|---|
| Circle | $x^2+y^2=r^2$ | $xx_1+yy_1=r^2$ | $y=mx\pm r\sqrt{1+m^2}$ |
| Parabola | $y^2=4ax$ | $yy_1=2a(x+x_1)$ | $y=mx+\frac{a}{m}$ |
| Ellipse | $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ | $\frac{xx_1}{a^2}+\frac{yy_1}{b^2}=1$ | $y=mx\pm\sqrt{a^2m^2+b^2}$ |
| Hyperbola | $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ | $\frac{xx_1}{a^2}-\frac{yy_1}{b^2}=1$ | $y=mx\pm\sqrt{a^2m^2-b^2}$ |
Related Topics
- Parabola - The Path of Projectiles
- Ellipse - The Oval of Orbits
- Hyperbola - The Difference Master
- General Conic Sections
- Chord of Contact and Pole-Polar
- Circles
- Straight Lines
Master the art of tangents and normals - where curves meet lines! 🎯