Parabola Class 11 Notes & Numericals for IIT JEE
eSaral Academic and Editorial Team
Summary
Parabola Class 11 Notes and numericals cover the definition of a parabola, the four standard equations, the parametric form (at², 2at), the position of a point, the tangent and the normal in point, slope and parametric forms, the chord of contact, the chord with a given midpoint, the pair of tangents, the focal chord and the reflection property.
The parabola is part of Chapter 10, Conic Sections, in the current NCERT Class 11 Maths book (Chapter 11 in older books). The CBSE board syllabus asks for the standard equations and the simple properties of a parabola. In the JEE Main and JEE Advanced syllabus, the parabola is part of coordinate geometry, and the problems on tangents, normals, chords and loci go much further. These Parabola Class 11 Notes cover both levels. Revise the earlier conics and lines in eSaral’s Circle notes and Straight Lines notes first.
Definition and Key Terms in Parabola Class 11 Notes
A parabola is the locus of a point $P$ whose distance from a fixed point (the focus $S$) is equal to its distance from a fixed line (the directrix), and the focus is not on the directrix:
$$PS = PM \quad (M \text{ is the foot of the perpendicular from } P \text{ to the directrix}), \qquad e = 1$$
The eccentricity of a parabola is always $e = 1$ (for an ellipse, $e < 1$, and for a hyperbola, $e > 1$).
Term | Meaning (for $y^2 = 4ax$) |
|---|---|
Focus $S$ | The fixed point $(a, 0)$ |
Directrix | The fixed line $x = -a$ |
Vertex | The midpoint of the perpendicular from the focus to the directrix, $(0, 0)$ |
Axis | The line of symmetry through the focus and the vertex ($y = 0$) |
Latus rectum | The focal chord perpendicular to the axis. Its length is $4a$, and its ends are $(a, \pm 2a)$ |
Focal distance of $P(x_1, y_1)$ | $PS = x_1 + a$ |
Focal chord | Any chord that passes through the focus |
Parabola Class 11 Notes & Numericals
India's Best Exam Preparation for Class 11th - Download Now
India's Best Exam Preparation for Class 11th - Download Now
India's Best Exam Preparation for Class 11th - Download Now
India's Best Exam Preparation for Class 11th - Download Now
India's Best Exam Preparation for Class 11th - Download Now
Standard Equations of a Parabola in Parabola Class 11 Notes
Equation ($a > 0$) | Opens | Focus | Directrix | Axis | Latus rectum |
|---|---|---|---|---|---|
$y^2 = 4ax$ | Right | $(a, 0)$ | $x = -a$ | $y = 0$ | $4a$ |
$y^2 = -4ax$ | Left | $(-a, 0)$ | $x = a$ | $y = 0$ | $4a$ |
$x^2 = 4ay$ | Up | $(0, a)$ | $y = -a$ | $x = 0$ | $4a$ |
$x^2 = -4ay$ | Down | $(0, -a)$ | $y = a$ | $x = 0$ | $4a$ |
With a shifted vertex $(h, k)$: $(y - k)^2 = 4a(x - h)$ has the focus $(h + a, k)$, the directrix $x = h - a$ and the axis $y = k$.
General second-degree equation: $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a parabola if $h^2 = ab$ and the curve is not degenerate, that is, $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 \neq 0$.
Parametric Form of the Parabola in Parabola Class 11 Notes
For $y^2 = 4ax$, every point can be written as $P(t) = (at^2, 2at)$. For $x^2 = 4ay$, it is $(2at, at^2)$. The parameter $t$ simplifies most of the results.
Result for $y^2 = 4ax$ | Formula |
|---|---|
Tangent at $P(t)$ | $ty = x + at^2$ |
Normal at $P(t)$ | $y + tx = 2at + at^3$ |
Slope of the tangent, slope of the normal | $\dfrac{1}{t}$, $-t$ |
Chord joining $P(t_1)$ and $P(t_2)$ | $(t_1 + t_2)y = 2x + 2at_1t_2$ |
Focal chord | $t_1 t_2 = -1$ |
The normal at $P(t)$ meets the parabola again at $Q(t')$ | $t' = -t - \dfrac{2}{t}$ |
Point of intersection of the tangents at $P(t_1)$ and $P(t_2)$ | $\big(at_1t_2,\ a(t_1 + t_2)\big)$ |
Position of a Point with Respect to a Parabola in Parabola Class 11 Notes
For $y^2 = 4ax$, the point $P(h, k)$ is inside if $k^2 - 4ah < 0$, on the parabola if $k^2 - 4ah = 0$, and outside if $k^2 - 4ah > 0$. (Inside means on the side of the focus.)
Tangent to a Parabola in Parabola Class 11 Notes
Form | Equation of the tangent to $y^2 = 4ax$ |
|---|---|
Point form (at $(x_1, y_1)$) | $yy_1 = 2a(x + x_1)$ |
Slope form (slope $m$) | $y = mx + \dfrac{a}{m}$, with the point of contact $\left(\dfrac{a}{m^2}, \dfrac{2a}{m}\right)$ |
Parametric form (at $t$) | $ty = x + at^2$ |
Condition of tangency: the line $y = mx + c$ touches $y^2 = 4ax$ if and only if $c = \dfrac{a}{m}$.
Properties of tangents:
- The tangent at $P(t)$ meets the axis at $T(-at^2, 0)$, and $SP = ST$ (so $\triangle SPT$ is isosceles).
- The foot of the perpendicular from the focus to any tangent lies on the tangent at the vertex ($x = 0$).
- The tangents at the ends of a focal chord are perpendicular, and they meet on the directrix. In general, the locus of the point from which two perpendicular tangents are drawn is the directrix.
- The tangent at the vertex is the $y$-axis ($x = 0$).
Normal to a Parabola in Parabola Class 11 Notes
Form | Equation of the normal to $y^2 = 4ax$ |
|---|---|
Point form (at $(x_1, y_1)$) | $y - y_1 = -\dfrac{y_1}{2a}(x - x_1)$ |
Slope form (slope $m$) | $y = mx - 2am - am^3$, with the foot $(am^2, -2am)$ |
Parametric form (at $t$) | $y + tx = 2at + at^3$ |
Properties of normals:
- The normal at $P(t)$ meets the axis at $G(2a + at^2, 0)$. The subnormal (the projection of $PG$ on the axis) is always $2a$.
- It meets the parabola again at the point with the parameter $t' = -t - \dfrac{2}{t}$.
- Three normals can be drawn from a point $(h, k)$. Their slopes are the roots of $am^3 + (2a - h)m + k = 0$, so $m_1 + m_2 + m_3 = 0$, $\ m_1m_2 + m_2m_3 + m_3m_1 = \dfrac{2a - h}{a}$ and $m_1m_2m_3 = -\dfrac{k}{a}$. In terms of the parameters, $t_1 + t_2 + t_3 = 0$ (the feet are “conormal” points). Three real normals exist when $h > 2a$ (and the discriminant condition is satisfied).
Chord of Contact, Midpoint Chord and Pair of Tangents in Parabola Class 11 Notes
Write $S = y^2 - 4ax$, $\ S_1 = y_1^2 - 4ax_1$ and $T = yy_1 - 2a(x + x_1)$.
Result | Equation |
|---|---|
Chord of contact (or polar) of $P(h, k)$ | $T = 0$, that is, $ky = 2a(x + h)$ |
Chord with the midpoint $(h, k)$ | $T = S_1$, that is, $k(y - k) = 2a(x - h)$ |
Pair of tangents from $(x_1, y_1)$ | $SS_1 = T^2$ |
Diameter (the midpoints of the parallel chords of slope $m$) | $y = \dfrac{2a}{m}$ |
Focal Chord and Reflection Property in Parabola Class 11 Notes
For a focal chord with the ends $P(t_1)$ and $Q(t_2)$ on $y^2 = 4ax$:
$$t_1 t_2 = -1, \qquad PQ = a\left(t + \frac{1}{t}\right)^2, \qquad \frac{1}{SP} + \frac{1}{SQ} = \frac{1}{a}$$
The shortest focal chord is the latus rectum ($t = 1$, length $4a$). A circle that has a focal chord as a diameter touches the directrix.
Reflection property: a ray parallel to the axis is reflected through the focus. This is why satellite dishes and the reflectors of telescopes are parabolic, and why a headlight puts its lamp at the focus.
Key Formulas for y² = 4ax in Parabola Class 11 Notes
Concept | Formula |
|---|---|
Eccentricity | $e = 1$ |
Latus rectum | $4a$ |
Focal distance | $x_1 + a$ |
Point on the parabola | $(at^2, 2at)$ |
Tangent (slope form) | $y = mx + a/m$ |
Normal (parametric) | $y + tx = 2at + at^3$ |
Chord of contact | $ky = 2a(x + h)$ |
Midpoint chord | $T = S_1$ |
Pair of tangents | $SS_1 = T^2$ |
Focal chord | $t_1t_2 = -1$ |
IIT JEE Methods and Tips in Parabola Class 11 Notes
- Use the parametric form $(at^2, 2at)$ for tangents, normals, chords and loci, because one parameter replaces $x$ and $y$.
- For a locus, take a general point, write the condition in terms of $t$ (or $m$), and eliminate the parameter.
- For a chord through the focus, write $t_1t_2 = -1$ at once.
- For tangents from a point, use the slope form $y = mx + a/m$, and find the two values of $m$ from the point.
- Check the orientation: for $x^2 = 4ay$, swap the roles of $x$ and $y$ (the parametric point is $(2at, at^2)$).
- Do not confuse the focal distance $x_1 + a$ with the length of the latus rectum $4a$.
Solved Examples from Parabola Class 11 Notes
Example 1 (the parts of a parabola): For $y^2 = 16x$, find the focus, the directrix and the length of the latus rectum.
$4a = 16$, so $a = 4$. The focus is $(4, 0)$, the directrix is $x = -4$, and the latus rectum is $4a = 16$.
Example 2 (the equation from the vertex and the focus): Find the parabola with the vertex $(1, 2)$ and the focus $(3, 2)$.
The axis is $y = 2$, and $a = 3 - 1 = 2$, opening to the right. So $(y - 2)^2 = 8(x - 1)$.
Example 3 (parametric point): Find the point with $t = 2$ on $y^2 = 8x$.
$a = 2$, so the point is $(at^2, 2at) = (8, 8)$. Check: $8^2 = 64 = 8\times 8$.
Example 4 (tangent with a given slope): Find the tangent to $y^2 = 8x$ with the slope 2, and the point of contact.
$a = 2$, $m = 2$, so $y = 2x + \dfrac{2}{2} = 2x + 1$. The point of contact is $\left(\dfrac{a}{m^2}, \dfrac{2a}{m}\right) = \left(\dfrac{1}{2}, 2\right)$.
Example 5 (condition of tangency): For what value of $c$ is $y = x + c$ tangent to $y^2 = 12x$?
$a = 3$ and $m = 1$, so $c = \dfrac{a}{m} = 3$.
Example 6 (normal): Find the normal at $t = 1$ to $y^2 = 4x$, and the point where it meets the parabola again.
$a = 1$, so the normal is $y + x = 2 + 1$, that is, $x + y = 3$. The other point has $t' = -1 - \dfrac{2}{1} = -3$, which gives $(9, -6)$. Check: $(-6)^2 = 36 = 4\times 9$, and $9 - 6 = 3$.
Example 7 (focal chord): One end of a focal chord of $y^2 = 4x$ is $(4, 4)$. Find the other end and the length of the chord.
$(4, 4)$ has $t = 2$, so the other end has $t_2 = -\dfrac{1}{2}$, which gives $\left(\dfrac{1}{4}, -1\right)$. The length is $a\left(t + \dfrac{1}{t}\right)^2 = \left(2 + \dfrac{1}{2}\right)^2 = \dfrac{25}{4}$. Check: $SP + SQ = (4 + 1) + \left(\dfrac{1}{4} + 1\right) = \dfrac{25}{4}$.
Example 8 (chord of contact): Find the chord of contact of the tangents from $(-2, 3)$ to $y^2 = 4x$.
$ky = 2a(x + h)$ gives $3y = 2(x - 2)$, that is, $2x - 3y - 4 = 0$.
Example 9 (midpoint chord): Find the chord of $y^2 = 8x$ with the midpoint $(2, 1)$.
$a = 2$, and $k(y - k) = 2a(x - h)$ gives $y - 1 = 4(x - 2)$, that is, $4x - y - 7 = 0$.
Example 10 (tangents at the ends of a focal chord): Show that the tangents at the ends of the focal chord of Example 7 meet on the directrix.
The tangents at $t_1$ and $t_2$ meet at $(at_1t_2, a(t_1 + t_2))$. Here $t_1t_2 = -1$, so $x = -1$, which is the directrix $x = -a$. The point is $\left(-1, \dfrac{3}{2}\right)$.
Example 11 (position of a point): Where are $(2, 3)$ and $(1, 1)$ with respect to $y^2 = 4x$?
For $(2, 3)$: $k^2 - 4ah = 9 - 8 = 1 > 0$, so it is outside. For $(1, 1)$: $1 - 4 = -3 < 0$, so it is inside.
Example 12 (a locus): Find the locus of the midpoints of the focal chords of $y^2 = 4ax$.
For the midpoint $(h, k)$, the chord is $T = S_1$, which is $k(y - k) = 2a(x - h)$. It passes through the focus $(a, 0)$, so $-k^2 = 2a(a - h)$, which gives $k^2 = 2a(h - a)$. The locus is the parabola $y^2 = 2a(x - a)$.
Common Mistakes in Parabola Class 11 Notes Problems
Mistake | Correct approach |
|---|---|
Writing the product of the slopes of three normals as $k/a$ | It is $-k/a$ |
Using $b^2 - 4ac = 0$ alone as the test for a parabola | Also check that the conic is not degenerate ($\Delta \neq 0$) |
Forgetting the sign in the focus of $y^2 = -4ax$ | The focus is $(-a, 0)$ |
Using the focal distance $x_1 + a$ for $x^2 = 4ay$ | There, the focal distance is $y_1 + a$ |
Taking $t_1 t_2 = 1$ for a focal chord | It is $t_1t_2 = -1$ |
Using the slope form of the tangent for a vertical tangent | A vertical tangent ($x = 0$) has no slope form |
Treating $(a, 2a)$ as the end of the focal chord for every case | It is the end of the latus rectum only |
Using $T = S_1$ for the chord of contact | $T = 0$ is the chord of contact, and $T = S_1$ is the chord with a given midpoint |
How to Study Parabola Class 11 Notes
- Read the NCERT chapter first, and download the textbook from eSaral’s NCERT books for Class 11 page.
- Make $y^2 = 4ax$ your reference form. Write from memory its focus, directrix, latus rectum, focal distance and parametric point, and then learn the other three orientations.
- Derive the tangent and the normal in the three forms yourself, and then practise one problem each on the tangent condition, the normal that meets the parabola again, the focal chord, the chord of contact and a locus.
- Revise with eSaral’s Class 11 Maths notes and, for the formulas of the other chapters, the trigonometry formulas for Class 11.
- Solve the problems of previous years with eSaral’s JEE Main chapter-wise PYQs and JEE Advanced chapter-wise PYQs.
Frequently Asked Questions
What are the important formulas of Parabola for JEE?
For $y^2 = 4ax$: the parametric point $(at^2, 2at)$, the focal distance $x_1 + a$, the latus rectum $4a$, the tangent $ty = x + at^2$ (or $y = mx + a/m$ with the condition $c = a/m$), the normal $y + tx = 2at + at^3$, the second meeting point of a normal $t' = -t - 2/t$, the focal chord condition $t_1t_2 = -1$, the chord of contact $ky = 2a(x + h)$ and the pair of tangents $SS_1 = T^2$. eSaral’s Parabola Class 11 Notes list all of them in one table.
How to study Parabola Class 11 for IIT JEE?
Start with $y^2 = 4ax$ as the reference form, learn the other three orientations, and derive the tangent and the normal in all three forms. Then practise the five problem types (tangent condition, normal rejoining, focal chord, chord of contact and locus), and solve previous year questions chapter-wise. These Parabola Class 11 Notes follow this order.
What are the standard equations of a parabola?
The four standard equations are $y^2 = 4ax$ (opens right, focus $(a, 0)$), $y^2 = -4ax$ (left, focus $(-a, 0)$), $x^2 = 4ay$ (up, focus $(0, a)$) and $x^2 = -4ay$ (down, focus $(0, -a)$). All have the vertex at the origin and the latus rectum $4a$.
What is the condition for a line to be tangent to y² = 4ax?
The line $y = mx + c$ touches $y^2 = 4ax$ if and only if $c = a/m$. It follows from substituting the line in the parabola and putting the discriminant equal to zero. The point of contact is $(a/m^2, 2a/m)$.
What is the significance of t₁t₂ = −1 for a focal chord?
A chord of $y^2 = 4ax$ with the ends $P(t_1)$ and $Q(t_2)$ passes through the focus if and only if $t_1t_2 = -1$. It also shows that the tangents at the ends of a focal chord meet on the directrix. In Parabola Class 11 Notes, it is the first result to write in every focal chord problem.
How do you find the equation of a chord with a given midpoint?
For $y^2 = 4ax$, the chord with the midpoint $(h, k)$ is $T = S_1$, that is, $k(y - k) = 2a(x - h)$. For example, the midpoint $(2, 1)$ on $y^2 = 8x$ gives $4x - y - 7 = 0$.
How many normals can be drawn from a point to a parabola?
In general, three normals can be drawn from a point $(h, k)$ to $y^2 = 4ax$. Their slopes are the roots of $am^3 + (2a - h)m + k = 0$, so the sum of the slopes is zero. Three real normals exist when $h > 2a$ (and the discriminant condition holds).
What is the reflection property of a parabola?
Any ray that is parallel to the axis passes through the focus after reflection from the parabola. This is why satellite dishes, telescope mirrors and headlight reflectors have a parabolic shape.
Is the parabola important for IIT JEE?
The parabola is part of coordinate geometry in the JEE Main and JEE Advanced syllabus, together with the circle, the ellipse and the hyperbola. The number of questions changes every year, so check previous years’ papers. The tangent, the normal, the focal chord and the locus problems are the common types, and these Parabola Class 11 Notes cover each of them.
Which chapter number is the parabola in Class 11 Maths?
In the current NCERT Class 11 Maths book, the parabola is in Chapter 10, Conic Sections (Chapter 11 in older books). eSaral has chapter-wise Class 11 Maths notes, NCERT books and JEE Main PYQs, and all of them are linked on this page.
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eSaral Academic and Editorial Team
Team eSaral is the collective author profile for educational content created by eSaral’s teachers and academic contributors. The team draws on expertise from IIT graduates, doctors, experienced educators and subject specialists to develop resources for JEE, NEET and school students. Our articles aim to explain concepts clearly and help students study with confidence.
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