Conic Sections — JEE Mains Mathematics

Formulas (6)

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

Watch out for (9)

• Divide by the $x^2$ coefficient first
  → The general equation: centre, radius and when it is a circle
• The centre is $(-g,-f)$, with the signs flipped
  → The general equation: centre, radius and when it is a circle
• For a point inside, the nearest distance is $r-PC$
  → Position of a point, and nearest and farthest distances
• Both points must be ends of ONE diameter
  → The diameter form and right angles
• Touching the $x$-axis fixes $r=|k|$, not $|h|$
  → Touching the axes and cutting intercepts
• Two parallel tangents give the DIAMETER, not the radius
  → Building a circle from conditions
• A distance condition gives two signs
  → Building a circle from conditions
• The radius scales by the MOVING end's share
  → Loci that turn out to be circles
• Equal distances give a line, not a circle
  → Loci that turn out to be circles

Formulas (5)

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

Watch out for (6)

• Use the distance from the CENTRE, not from the origin
  → Chord length from the distance to the centre
• $T=S_1$ is not $T=0$
  → Chords by their midpoint, or by the angle they subtend
• The slope form needs the circle centred at the origin
  → When a line is a tangent, and the tangent at a point
• The angle is TWICE $\tan^{-1}\frac{r}{L}$
  → Tangents from an outside point: length, chord of contact, angle, area
• Triangle $PAB$ is not triangle $PAC$
  → Tangents from an outside point: length, chord of contact, angle, area
• Find the centre before measuring
  → Parametric points and largest and smallest values

Formulas (4)

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

Watch out for (4)

• Two points needs BOTH inequalities
  → Relative position and the number of common tangents
• Make the $x^2$ coefficients equal before subtracting
  → The common chord, and a diameter that is a chord of another circle
• $\lambda=-1$ is not a circle
  → Circles through the meeting points of two curves
• Equal radii give a second equation
  → The image of a circle in a line

Formulas (6)

• 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
  $$P=(at^2,\,2at)$$

Watch out for (6)

• Shift the focus along with the vertex
  → Standard and shifted forms: vertex, focus, directrix, latus rectum
• Keep the $\sqrt{l^2+m^2}$
  → Parabolas in any position: the focus-directrix definition
• $-4$ is for the vertex, $-1$ is for the focus
  → The parametric point and chords seen from the vertex
• $\theta$ is the angle with the AXIS
  → Focal chords and focal distances
• Use the MIDPOINT's ordinate
  → Chords of a parabola by their midpoint, and where a line meets it
• Read the NEW curve's features
  → Loci from a moving point on a parabola

Formulas (3)

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

Watch out for (3)

• The contact point is $\left(\frac{a}{m^2},\frac{2a}{m}\right)$, not $(a,2a)$
  → The tangent: point, parametric and slope forms
• Shift before using $x=-a$
  → Two tangents: from a point, where they meet, and the directrix
• The slope-form foot is $(am^2,-2am)$
  → The normal, and the shortest distance to a parabola

Formulas (4)

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

Watch out for (4)

• $1-e^2$ for an ellipse, $e^2-1$ for a hyperbola
  → a, b, e and the latus rectum
• Check which denominator is bigger
  → A vertical major axis, or a moved centre
• $a-ex$ is the distance to the focus on the SAME side
  → Focal distances: the constant sum
• The auxiliary circle has radius $a$, the SEMI-MAJOR axis
  → The parametric point, the auxiliary circle and loci

Formulas (4)

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

Watch out for (4)

• Shift the line before using $c^2=a^2m^2+b^2$
  → The tangent: point, parametric and slope forms
• The director circle uses $a^2+b^2$, not $a^2$
  → Pairs of tangents, the chord of contact and the director circle
• The normal has a MINUS between its terms
  → The normal to an ellipse
• $T=S_1$ has $S_1$ on the right, not $1$
  → Chords: by midpoint, through a point, and at right angles at the centre

Formulas (4)

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

Watch out for (4)

• $e^2-1$, not $1-e^2$
  → a, b, e, the latus rectum and the conjugate hyperbola
• Do not reuse $a^2-b^2$ for the hyperbola
  → Ellipse and hyperbola together: shared foci and linked eccentricities
• The DIFFERENCE is $2a$, not the sum
  → Focal distances: the constant difference, and foci anywhere
• $xy=c^2$ has its axes along $y=\pm x$
  → The rectangular hyperbola, and hyperbolas that appear as loci

Formulas (2)

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

Watch out for (2)

• Minus $b^2$ for the hyperbola
  → Tangents, and when a line meets a hyperbola
• PLUS in the normal, MINUS in the tangent
  → The normal to a hyperbola

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