JEE Mains Maths · Conic Sections

Common Tangents and Loci Across Conics

Questions that join two different curves: a line touching both, a tangent to one curve tested against another, loci of midpoints of chords of one curve that touch another, and the angle at which two curves cross.

All Conic Sections notes JEE Mains Maths strategy

Why this matters

Thirty-one PYQs, and they carry no new formula. Each one applies the slope-form tangency conditions from the other pages twice, once per curve. The skill is keeping the table of conditions straight. Four ideas cover the page.

Concept 1 of 4: Common tangents: one line, two tangency conditions

Write the line in the slope form that already touches the first curve, so its intercept is a function of

m

. Then impose the second curve's tangency condition on that same line. You get one equation in

m

; its roots are the slopes of the common tangents. Everything depends on having the five conditions ready.

Definition

For the line $y=mx+c$ :

Circle $x^2+y^2=r^2$ : $c^2=r^2(1+m^2)$ .
Parabola $y^2=4ax$ : $c=\frac{a}{m}$ . Parabola $x^2=4ay$ : $c=-am^2$ .
Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ : $c^2=a^2m^2+b^2$ .
Hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ : $c^2=a^2m^2-b^2$ .
For a shifted curve, move the origin to its centre or vertex first.

Parabola and circle, both about the origin

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

Worked example

Find the common tangents of

y^2=8x

and

x^2+y^2=2

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 27 July 2022 · Q179Moderate

Example 1 · Conic Sections · Common Tangents and Loci Across Conics

A common tangent

T

to the curves

C_{1}:\frac{x^{2}}{4}+\frac{y^{2}}{9}= 1

and

C_{2}:\frac{x^{2}}{42}-\frac{y^{2}}{143}= 1

does not pass through the fourth quadrant. If

T

touches

C_{1}

\left( x_{1},y_{1} \right)

and

C_{2}

\left( x_{2},y_{2} \right)

, then

\left| 2x_{1}+x_{2} \right|

is equal to

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.

Drill 14 more on common tangents: one line, two tangency conditions

Concept 2 of 4: A tangent to one curve, tested on another

Many questions build a tangent to one curve, usually at a given point, and then ask about it against a second curve: does it touch it, where does it cut it, what triangle does it make. Work in two clean steps. First write the tangent from the first curve's rules. Then treat it as an ordinary line for the second curve.

Definition

Step 1: the tangent to the first curve (point form, parametric form or slope form).
Step 2: for the second curve use the distance test (circle), the tangency condition (conic) or substitution (for points of intersection).
A circle touching a conic at a point has its centre on the conic's normal there.

Tangent to y² = 4ax at (x₁, y₁)

yy_1=2a(x+x_1)

Worked example

The tangent to

x^2=12y

(6,3)

: does it touch

x^2+y^2=\frac92

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · Paper 23 · Q85Moderate

Example 2 · Conic Sections · Common Tangents and Loci Across Conics

A tangent line

L

is drawn at the point

(2, - 4)

on the parabola

y^{2}= 8x

. If the line

L

is also tangent to the circle

x^{2}+y^{2}=a

, then 'a' is equal to

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.

Drill 7 more on a tangent to one curve, tested on another

Concept 3 of 4: Loci of midpoints of chords that touch another curve

Two tools combine. The chord of the first curve with midpoint

(h,k)

T=S_1

, a line whose slope and intercept depend on

h

and

k

. Requiring that line to touch the second curve gives one equation in

h

and

k

: that equation is the locus.

Definition

Write the chord with midpoint $(h,k)$ as $T=S_1$ .
Put it in the form $y=mx+c$ (or use the distance test for a circle).
Impose the second curve's tangency condition.
Rename $h,k$ as $x,y$ .

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

hx+ky=h^2+k^2

Worked example

Find the locus of the midpoints of chords of

x^2+y^2=9

that touch

y^2=4x

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · Paper 19 · Q72Moderate

Example 3 · Conic Sections · Common Tangents and Loci Across Conics

The locus of the mid points of the chords of the hyperbola

x^{2}-y^{2}= 4

, which touch the parabola

y^{2}= 8x

, is:

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

x,y

too early mixes the two and wrecks the algebra.

Drill 3 more on loci of midpoints of chords that touch another curve

Concept 4 of 4: The angle between two curves, and curves that cut at right angles

Two curves cross at the angle between their tangents at the crossing point. Find the point, find each slope (by differentiating implicitly), and use the angle formula for two lines. For two central conics there is a shortcut: they cut at right angles exactly when they share their foci.

Definition

At the meeting point, find $m_1$ and $m_2$ by implicit differentiation.
$\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|$ ; right angle when $m_1m_2=-1$ .
$\frac{x^2}{a}+\frac{y^2}{b}=1$ and $\frac{x^2}{c}+\frac{y^2}{d}=1$ cut at right angles when $a-b=c-d$ (same foci).
An ellipse and a hyperbola with the same foci always cut at right angles.

Angle between the curves

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

Worked example

Find the angle between

y^2=4x

and

x^2=4y

(4,4)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · Paper 23 · Q71Moderate

Example 4 · Conic Sections · Common Tangents and Loci Across Conics

The angle of intersection of the curves

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1

and

x^{2}+y^{2}=ab,\ a>b,

is:

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.

Drill 3 more on the angle between two curves, and curves that cut at right angles

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

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

Watch out for (4)

Equate the INTERCEPTS, with one slope
→ Common tangents: one line, two tangency conditions
Use the tangency condition of the SECOND curve
→ A tangent to one curve, tested on another
Keep $h,k$ as constants until the end
→ Loci of midpoints of chords that touch another curve
The angle uses slopes AT THE MEETING POINT
→ The angle between two curves, and curves that cut at right angles

Test yourself on Conic Sections

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Conic Sections chapter test Drill all 31 questions