Conic Sections — JEE Mains Maths Formulas and Traps

Equation of a Circle

The general equation: centre, radius and when it is a circle

Centre and radius of the general form

C=(-g,-f),\qquad r=\sqrt{g^2+f^2-c}

Position of a point, and nearest and farthest distances

Power of a point

S_1=x_1^2+y_1^2+2gx_1+2fy_1+c=PC^2-r^2

The diameter form and right angles

Diameter form

(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0

Touching the axes and cutting intercepts

Intercepts on the axes

\ell_x=2\sqrt{g^2-c},\qquad \ell_y=2\sqrt{f^2-c}

Building a circle from conditions

Distance from the centre to a tangent line

\frac{|ah+bk+c|}{\sqrt{a^2+b^2}}=r

Loci that turn out to be circles

A dividing point whose far end moves on a circle

P=\frac{nA+mB}{m+n}\ \Rightarrow\ \text{radius}=\frac{m}{m+n}\,r

Common traps

Divide by the $x^2$ coefficient first

In

2x^2+2y^2-8x+12y+6=0

, reading

2g=-8

directly gives the wrong centre

(4,-6)

. The formulas

(-g,-f)

and

\sqrt{g^2+f^2-c}

hold only after the

x^2

and

y^2

coefficients are

1

.

The centre is $(-g,-f)$ , with the signs flipped

x^2+y^2-4x+6y=0

has

2g=-4

, so

g=-2

and the centre's

x

-coordinate is

+2

. Half the coefficient, then change the sign.

For a point inside, the nearest distance is $r-PC$

Writing

PC-r

for an inside point gives a negative 'distance'. Use

|PC-r|

: it is

PC-r

outside and

r-PC

inside.

Both points must be ends of ONE diameter

The diameter form through two points that are merely ON the circle gives a different, smaller circle. First check that the segment really is a diameter, for example by a right angle standing on it.

Touching the $x$ -axis fixes $r=|k|$ , not $|h|$

The radius to the point of contact is perpendicular to the axis, so it runs vertically: its length is the centre's

y

-coordinate. Mixing up

h

and

k

here is the usual slip.

Two parallel tangents give the DIAMETER, not the radius

The gap between two parallel tangents spans the whole circle, so

r

is half of it. Taking the gap as

r

doubles the radius.

A distance condition gives two signs

|ah+bk+c|=r\sqrt{a^2+b^2}

splits into two cases. Keep both until a stated condition (a quadrant, 'below the axis') rules one out.

The radius scales by the MOVING end's share

If

P

divides

AB

as

m:n

from the fixed point

A

,

P

is

\frac{m}{m+n}

of the way to

B

, so its circle has radius

\frac{m}{m+n}r

. Using

\frac{n}{m+n}

swaps the shares.

Equal distances give a line, not a circle

The ratio rule gives a circle only for

\lambda\neq1

. When

PA=PB

, the squared terms cancel and the locus is the perpendicular bisector.

Chords and Tangents of a Circle

Learn this subtopic in the notes

Chord length from the distance to the centre

Chord length

\ell=2\sqrt{r^2-d^2}

Chords by their midpoint, or by the angle they subtend

Chord with a given midpoint

T=S_1

When a line is a tangent, and the tangent at a point

Tangent of slope m to x² + y² = a²

y=mx\pm a\sqrt{1+m^2}

Tangents from an outside point: length, chord of contact, angle, area

Tangent length and area of triangle PAB

L=\sqrt{S_1},\qquad [PAB]=\frac{rL^3}{r^2+L^2}

Parametric points and largest and smallest values

Parametric point

(h+r\cos\theta,\ k+r\sin\theta)

Common traps

Use the distance from the CENTRE, not from the origin

For a circle not centred at the origin, the chord length needs the perpendicular distance from

(-g,-f)

to the line. Measuring from

(0,0)

out of habit gives a wrong

d

.

$T=S_1$ is not $T=0$

T=0

is the tangent (or polar) at

(x_1,y_1)

. The chord with midpoint

(x_1,y_1)

is

T=S_1

. Mixing them gives a line through the wrong point.

The slope form needs the circle centred at the origin

y=mx\pm a\sqrt{1+m^2}

is for

x^2+y^2=a^2

. For any other circle, use the distance test from the actual centre, or shift the origin first.

The angle is TWICE $\tan^{-1}\frac{r}{L}$

\tan^{-1}\frac{r}{L}

is the angle between one tangent and

PC

. The angle between the two tangents is double it.

Triangle $PAB$ is not triangle $PAC$

[PAC]=\frac12rL

is half the quadrilateral

PACB

. The triangle cut off by the chord of contact is smaller:

\frac{rL^3}{r^2+L^2}

.

Find the centre before measuring

Extremes of distance run along the line through the CENTRE. Working from a point on the circle that merely looks nearest gives the wrong value.

Two Circles and Families of Circles

Learn this subtopic in the notes

Relative position and the number of common tangents

Two circles meet in two points when

|r_1-r_2|<C_1C_2<r_1+r_2

The common chord, and a diameter that is a chord of another circle

Common chord

S_1-S_2=0

Circles through the meeting points of two curves

Family through a circle and a line

S+\lambda L=0

The image of a circle in a line

Reflection of a point in a line

(x',y')=(x_0,y_0)-\frac{2(ax_0+by_0+c)}{a^2+b^2}\,(a,b)

Common traps

Two points needs BOTH inequalities

d<r_1+r_2

alone allows one circle to sit inside the other. Two intersection points needs

|r_1-r_2|<d

as well.

Make the $x^2$ coefficients equal before subtracting

S_1-S_2

gives a line only when both equations have

x^2+y^2

with coefficient

1

. Divide out first, or the squared terms survive.

$\lambda=-1$ is not a circle

In

S_1+\lambda S_2=0

the

x^2+y^2

terms cancel at

\lambda=-1

, leaving the common chord. Exclude it when a circle is wanted.

Equal radii give a second equation

When the image circle is given with unknown coefficients, the equal radii are a condition too:

r_1^2=r_2^2

often fixes the last constant. Reflecting only the centre and stopping leaves it unused.

Parabola and Its Focal Chords

Learn this subtopic in the notes

Standard and shifted forms: vertex, focus, directrix, latus rectum

The standard parabola

y^2=4ax:\quad S=(a,0),\ \ x=-a,\ \ \text{LR}=4a

Parabolas in any position: the focus-directrix definition

Focus-directrix equation

(x-\alpha)^2+(y-\beta)^2=\frac{(lx+my+n)^2}{l^2+m^2}

The parametric point and chords seen from the vertex

Chord joining t₁ and t₂

2x-(t_1+t_2)y+2at_1t_2=0

Focal chords and focal distances

Focal chord at angle θ to the axis

t_1t_2=-1,\qquad PQ=\frac{4a}{\sin^2\theta}

Chords of a parabola by their midpoint, and where a line meets it

Slope of a chord of y² = 4ax with midpoint (h, k)

m=\frac{2a}{k}

Loci from a moving point on a parabola

Parametric point to eliminate

P=(at^2,\,2at)

Common traps

Shift the focus along with the vertex

For

(y-k)^2=4a(x-h)

the focus is

(h+a,k)

, not

(a,0)

. Every feature moves with the vertex.

Keep the $\sqrt{l^2+m^2}$

The distance to the directrix is

\frac{|lx+my+n|}{\sqrt{l^2+m^2}}

. Dropping the denominator gives a different curve.

$-4$ is for the vertex, $-1$ is for the focus

A right angle at the vertex gives

t_1t_2=-4

. A chord through the focus gives

t_1t_2=-1

. Swapping them is the most common slip on this page.

$\theta$ is the angle with the AXIS

\frac{4a}{\sin^2\theta}

uses the chord's angle with the parabola's axis. For

x^2=4ay

the axis is vertical, so measure from the

y

-axis.

Use the MIDPOINT's ordinate

The slope

\frac{2a}{k}

uses the

y

-coordinate of the midpoint. Plugging in an end's

y

gives the tangent's slope at that end instead.

Read the NEW curve's features

After finding the locus, questions ask for its latus rectum or directrix. Rewrite it in standard form first; the original parabola's

a

no longer applies.

Tangents and Normals to a Parabola

Learn this subtopic in the notes

The tangent: point, parametric and slope forms

Slope form for y² = 4ax

y=mx+\frac{a}{m},\qquad \text{touching at }\left(\frac{a}{m^2},\frac{2a}{m}\right)

Two tangents: from a point, where they meet, and the directrix

Where the tangents at t₁ and t₂ meet

(at_1t_2,\ a(t_1+t_2))

The normal, and the shortest distance to a parabola

Normal in slope form, y² = 4ax

y=mx-2am-am^3,\qquad \text{foot }(am^2,\,-2am)

Common traps

The contact point is $\left(\frac{a}{m^2},\frac{2a}{m}\right)$ , not $(a,2a)$

(a,2a)

is the end of the latus rectum, where the slope is

1

. For any other slope use

\left(\frac{a}{m^2},\frac{2a}{m}\right)

.

Shift before using $x=-a$

For

y^2=16(x-3)

the vertex is at

(3,0)

, so the directrix is

x=3-4=-1

, not

x=-4

.

The slope-form foot is $(am^2,-2am)$

For the normal

y=mx-2am-am^3

the foot has a MINUS

2am

. The tangent's contact point

\left(\frac{a}{m^2},\frac{2a}{m}\right)

is a different point with a different

m

.

Ellipse: Axes, Eccentricity and Focal Distances

Learn this subtopic in the notes

a, b, e and the latus rectum

The linking relation and the latus rectum

b^2=a^2(1-e^2),\qquad \text{LR}=\frac{2b^2}{a}

A vertical major axis, or a moved centre

Tall ellipse (b > a)

a^2=b^2(1-e^2),\quad S=(0,\pm be),\quad \text{LR}=\frac{2a^2}{b}

Focal distances: the constant sum

Focal distances

SP=a-ex_1,\quad S'P=a+ex_1,\quad SP+S'P=2a

The parametric point, the auxiliary circle and loci

Parametric point and area

(a\cos\theta,\ b\sin\theta),\qquad \text{Area}=\pi ab

Common traps

$1-e^2$ for an ellipse, $e^2-1$ for a hyperbola

For an ellipse

b^2=a^2(1-e^2)

and

e<1

. Writing

a^2(e^2-1)

, the hyperbola rule, makes

b^2

negative.

Check which denominator is bigger

For

\frac{x^2}{9}+\frac{y^2}{25}=1

, using

b^2=a^2(1-e^2)

with

a^2=9

gives

1-e^2>1

, which is impossible. The larger denominator always plays the semi-major role.

$a-ex$ is the distance to the focus on the SAME side

For

S=(ae,0)

,

SP=a-ex_1

. The focus at

(-ae,0)

gives

a+ex_1

. Swapping them matters when only one focal distance is asked.

The auxiliary circle has radius $a$ , the SEMI-MAJOR axis

For a tall ellipse (

b>a

) the auxiliary circle is

x^2+y^2=b^2

. Always use the larger semi-axis.

Tangents, Normals and Chords of an Ellipse

Learn this subtopic in the notes

The tangent: point, parametric and slope forms

Tangency condition

y=mx+c\ \text{touches}\ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\iff c^2=a^2m^2+b^2

Pairs of tangents, the chord of contact and the director circle

Director circle

x^2+y^2=a^2+b^2

The normal to an ellipse

Normal at the parametric point

ax\sec\theta-by\csc\theta=a^2-b^2

Chords: by midpoint, through a point, and at right angles at the centre

Chord with midpoint (h, k)

\frac{hx}{a^2}+\frac{ky}{b^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}

Common traps

Shift the line before using $c^2=a^2m^2+b^2$

The condition is for an ellipse centred at the origin. For

\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1

, rewrite the line in

X=x-h

,

Y=y-k

first; the slope stays, the intercept changes.

The director circle uses $a^2+b^2$ , not $a^2$

x^2+y^2=a^2

is the auxiliary circle, the circle on the major axis. The meeting points of perpendicular tangents are farther out, at radius

\sqrt{a^2+b^2}

.

The normal has a MINUS between its terms

The tangent is

\frac{xx_1}{a^2}+\frac{yy_1}{b^2}=1

; the normal is

\frac{a^2x}{x_1}-\frac{b^2y}{y_1}=a^2-b^2

. The coordinates move to the denominators and the sign changes.

$T=S_1$ has $S_1$ on the right, not $1$

The chord with midpoint

(h,k)

ends in

\frac{h^2}{a^2}+\frac{k^2}{b^2}

. Writing

1

there gives the tangent-like line

T=0

, which passes through a different point.

Hyperbola: Axes, Eccentricity and Focal Distances

Learn this subtopic in the notes

a, b, e, the latus rectum and the conjugate hyperbola

The linking relation

b^2=a^2(e^2-1),\qquad \text{LR}=\frac{2b^2}{a}

Ellipse and hyperbola together: shared foci and linked eccentricities

The focal distance, two ways

c^2=a^2-b^2\ \text{(ellipse)},\qquad c^2=A^2+B^2\ \text{(hyperbola)}

Focal distances: the constant difference, and foci anywhere

Focal distances on the right branch

SP=ex-a,\quad S'P=ex+a,\quad S'P-SP=2a

The rectangular hyperbola, and hyperbolas that appear as loci

Rectangular hyperbola

xy=c^2:\ \left(ct,\frac{c}{t}\right),\qquad e=\sqrt2

Common traps

$e^2-1$ , not $1-e^2$

For a hyperbola

b^2=a^2(e^2-1)

and

e>1

. A root

e<1

of a given equation belongs to an ellipse, not to this curve.

Do not reuse $a^2-b^2$ for the hyperbola

The ellipse gives

c^2=a^2-b^2

; the hyperbola needs

c^2=A^2+B^2

. Using the ellipse rule on the hyperbola gives a negative or wrong

B^2

.

The DIFFERENCE is $2a$ , not the sum

For an ellipse

SP+S'P=2a

; for a hyperbola it is

|SP-S'P|=2a

. A question giving the SUM of focal distances on a hyperbola is giving

2ex

, not

2a

.

$xy=c^2$ has its axes along $y=\pm x$

The vertices of

xy=c^2

are

(c,c)

and

(-c,-c)

, not on the coordinate axes. Its transverse axis is the line

y=x

.

Tangents and Normals to a Hyperbola

Learn this subtopic in the notes

Tangents, and when a line meets a hyperbola

Tangency condition

y=mx+c\ \text{touches}\ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\iff c^2=a^2m^2-b^2

The normal to a hyperbola

Normal at (x₁, y₁)

\frac{a^2x}{x_1}+\frac{b^2y}{y_1}=a^2+b^2

Common traps

Minus $b^2$ for the hyperbola

For the ellipse

c^2=a^2m^2+b^2

; for the hyperbola

c^2=a^2m^2-b^2

. A slope with

a^2m^2<b^2

gives no tangent at all.

PLUS in the normal, MINUS in the tangent

The tangent is

\frac{xx_1}{a^2}-\frac{yy_1}{b^2}=1

and the normal is

\frac{a^2x}{x_1}+\frac{b^2y}{y_1}=a^2+b^2

. The signs flip between them.

Common Tangents and Loci Across Conics

Learn this subtopic in the notes

Common tangents: one line, two tangency conditions

Parabola and circle, both about the origin

c=\frac{a}{m}\ \ \text{and}\ \ c^2=r^2(1+m^2)

A tangent to one curve, tested on another

Tangent to y² = 4ax at (x₁, y₁)

yy_1=2a(x+x_1)

Loci of midpoints of chords that touch another curve

Chord of x² + y² = r² with midpoint (h, k)

hx+ky=h^2+k^2

The angle between two curves, and curves that cut at right angles

Angle between the curves

\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|

Common traps

Equate the INTERCEPTS, with one slope

Both conditions must be about the same line

y=mx+c

. Writing one curve's tangent with slope

m

and the other's with a fresh slope, then matching, loses the fact that it is one line.

Use the tangency condition of the SECOND curve

After step 1 the line is fixed. Checking it against the first curve's condition again only confirms step 1; the question is about the second curve.

Keep $h,k$ as constants until the end

Inside

T=S_1

,

x

and

y

are the line's running coordinates and

h,k

are fixed. Renaming

h,k

as

x,y

too early mixes the two and wrecks the algebra.

The angle uses slopes AT THE MEETING POINT

Find the intersection first. The slopes change along each curve, so a slope taken anywhere else gives a meaningless angle.

Conic Sections formulas

Equation of a Circle

The general equation: centre, radius and when it is a circle

Position of a point, and nearest and farthest distances

The diameter form and right angles

Touching the axes and cutting intercepts

Building a circle from conditions

Loci that turn out to be circles

Common traps

Chords and Tangents of a Circle

Chord length from the distance to the centre

Chords by their midpoint, or by the angle they subtend

When a line is a tangent, and the tangent at a point

Tangents from an outside point: length, chord of contact, angle, area

Parametric points and largest and smallest values

Common traps

Two Circles and Families of Circles

Relative position and the number of common tangents

The common chord, and a diameter that is a chord of another circle

Circles through the meeting points of two curves

The image of a circle in a line

Common traps

Parabola and Its Focal Chords

Standard and shifted forms: vertex, focus, directrix, latus rectum

Parabolas in any position: the focus-directrix definition

The parametric point and chords seen from the vertex

Focal chords and focal distances

Chords of a parabola by their midpoint, and where a line meets it

Loci from a moving point on a parabola

Common traps

Tangents and Normals to a Parabola

The tangent: point, parametric and slope forms

Two tangents: from a point, where they meet, and the directrix

The normal, and the shortest distance to a parabola

Common traps

Ellipse: Axes, Eccentricity and Focal Distances

a, b, e and the latus rectum

A vertical major axis, or a moved centre

Focal distances: the constant sum

The parametric point, the auxiliary circle and loci

Common traps

Tangents, Normals and Chords of an Ellipse

The tangent: point, parametric and slope forms

Pairs of tangents, the chord of contact and the director circle

The normal to an ellipse

Chords: by midpoint, through a point, and at right angles at the centre

Common traps

Hyperbola: Axes, Eccentricity and Focal Distances

a, b, e, the latus rectum and the conjugate hyperbola

Ellipse and hyperbola together: shared foci and linked eccentricities

Focal distances: the constant difference, and foci anywhere

The rectangular hyperbola, and hyperbolas that appear as loci

Common traps

Tangents and Normals to a Hyperbola

Tangents, and when a line meets a hyperbola

The normal to a hyperbola

Common traps

Common Tangents and Loci Across Conics

Common tangents: one line, two tangency conditions

A tangent to one curve, tested on another

Loci of midpoints of chords that touch another curve

The angle between two curves, and curves that cut at right angles

Common traps

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