Conic Sections Glossary - Complete

• Circle Show/Hide Details
  A set of points such that distance of each point from a fixed point (centre) remains constant.

  • Center Show/Hide Details
    $(-g, -f)$.
  • Radius Show/Hide Details
    The constant distance. $r=\sqrt{g^{2}+f^{2}-c}>0$ (Real circle). $r=\sqrt{g^{2}+f^{2}-c}=0$ (Point circle). If $g^{2}+f^{2}-c<0$ (Imaginary circle).
  • Standard Form of Equations of Circle Show/Hide Details
    Centre (h,k), radius = r: $(x-h)^{2}+(y-k)^{2}=\dot{r}^{2}$.
    Centre (0,0), radius = r: $x^{2}+y^{2}=r^{2}$.
    Centre (0,0), radius = 1 (Unit circle): $x^{2}+y^{2}=1$.
    Centre (0,0), radius = 0 (Point circle): $x^{2}+y^{2}=0$.
    If end points of the diameter are $A(x_{1},y_{1})\&B(x_{2},y_{2})$: $(x-x_{1})(x-x_{2})+(y-y_{1})(y-y_{2})=0$.
  • Parametric Form of Equations of Circle Show/Hide Details
    Standard form $(x-h)^{2}+(y-k)^{2}=r^{2}$: $x=h+r~cos~\theta,y=k+r~sin~\theta$.
    $x^{2}+y^{2}=r^{2}$: $x=r~cos~\theta,y=r~sin~\theta$.
  • General Form of a Circle Show/Hide Details
    $x^{2}+y^{2}+2gx+2fy+c=0$.

    • Constants: g, f, c.
    • Coefficients of $x^{2}$ and $y^{2}$ are 1.
    • Does not contain the term involving the product xy.
  • Tangents and Normals Show/Hide Details
    Tangent: A straight line that touches the curve at a point without cutting the curve.
    Normal: A straight line perpendicular to the curve at the point of tangency.

    • Equation of tangent to $x^{2}+y^{2}=a^{2}$ at $P(x_{1},y_{1})$: $xx_{1}+yy_{1}=a^{2}$.
    • Equation of normal to $x^{2}+y^{2}=a^{2}$ at $P(x_{1},y_{1})$: $xy_{1}-yx_{1}=0$.
    • Equation of tangent to $x^{2}+y^{2}+2gx+2fy+c=0$ at $P(x_{1},y_{1})$: $xx_{1}+yy_{1}+g(x+x_{1})+f(y+y_{1})+c=0$.
    • Equation of normal to $x^{2}+y^{2}+2gx+2fy+c=0$ at $P(x_{1},y_{1})$: $(x-x_{1})(y_{1}+f)-(y-y_{1})(x_{1}+g)=0$.
  • Condition of Tangency Show/Hide Details
    The line $y=mx+c$ touches the circle $x^{2}+y^{2}=a^{2}$ if $c=\pm a\sqrt{1+m^{2}}$.

    • Intersects in two distinct points if $a^{2}(1+m^{2})-c^{2}>0$.
    • Intersects in real and coincident points if $a^{2}(1+m^{2})-c^{2}=0$.
    • Intersects in imaginary points if $a^{2}(:+m^{2})-c^{2}<0$.< /li>
  • Equation of a Circle in Some Special Cases Show/Hide Details
    • \(|k| = r\)
    • Only one solution for \(x\) when \(y = 0\)
    • If the circle touches x-axis then its equation is:
      • Center (h,k): $(x-h)^{2}+(y-k)^{2}=k^{2} \Rightarrow x^{2}+y^{2}-2hx-2ky+h^{2}=0$.
      • Center (-h,k): $(x+h)^{2}+(y-k)^{2}=k^{2} \Rightarrow x^{2}+y^{2}+2hx-2ky+h^{2}=0$.
      • Center (-h,-k): $(x+h)^{2}+(y+k)^{2}=k^{2} \Rightarrow x^{2}+y^{2}+2hx+2ky+h^{2}=0$.
      • Center (h,-k): $(x-h)^{2}+(y+k)^{2}=k^{2} \Rightarrow x^{2}+y^{2}-2hx+2ky+h^{2}=0$.
    • \(|h| = r\)
    • Only one solution for \(y\) when \(x = 0\)
    • Equation of a circle touches y-axis then its equation is:
      • Center (h,k): $(x-h)^{2}+(y-k)^{2}=h^{2} \Rightarrow x^{2}+y^{2}-2hx-2ky+k^{2}=0$.
      • Center (-h,k): $(x+h)^{2}+(y-k)^{2}=h^{2} \Rightarrow x^{2}+y^{2}+2hx-2ky+k^{2}=0$.
      • Center (-h,-k): $(x+h)^{2}+(y+k)^{2}=h^{2} \Rightarrow x^{2}+y^{2}+2hx+2ky+k^{2}=0$.
      • Center (h,-k): $(x-h)^{2}+(y+k)^{2}=h^{2} \Rightarrow x^{2}+y^{2}-2hx+2ky+k^{2}=0$.
    • \(|h| = |k| = r\)
    • Equation of circle which touches both the axes is:
      • Center (r,r): $(x-r)^{2}+(y-r)^{2}=r^{2} \Rightarrow x^{2}+y^{2}-2rx-2ry+r^{2}=0$.
      • Center (-r,r): $(x+r)^{2}+(y-r)^{2}=r^{2} \Rightarrow x^{2}+y^{2}+2rx-2ry+r^{2}=0$.
      • Center (-r,-r): $(x+r)^{2}+(y+r)^{2}=r^{2} \Rightarrow x^{2}+y^{2}+2rx+2ry+r^{2}=0$.
      • Center (r,-r): $(x-r)^{2}+(y+r)^{2}=r^{2} \Rightarrow x^{2}+y^{2}-2rx+2ry+r^{2}=0$.
  • Important Properties of a Circle Show/Hide Details
    • Perpendicular dropped from the center of a circle on a chord bisects the chord.
    • The perpendicular bisector of any chord of a circle passes through the center of the circle.
    • The line joining the center of a circle to the midpoint of a chord is perpendicular to the chord.
    • Congruent chords of a circle are equidistant from its centre.
    • Measure of the central angle of a minor arc is double the measure of the angle subtended in the corresponding major arc.
    • An angle in a semi-circle is a right angle.
    • The tangent to a circle at any point of the circle is perpendicular to the radial segment at that point.
    • The perpendicular at the outer end of a radial segment is tangent to the circle.
    • Normal lines of a circle pass through the centre of the circle.
    • Mid point of the hypotenuse of a right triangle is the circumcentre of the triangle.
    • Perpendicular dropped from a point of a circle on a diameter is a mean proportional between the segments into which it divides the diameter.
  • Contact of Two Circles Show/Hide Details
    Let $r_{1}$ and $r_{2}$ be the radii of two circles and $C_{1}$, $C_{2}$ be their centers.

    Condition	Relationship
    Circles Touch Externally	$|C_{1}C_{2}|=r_{1}+r_{2}$
    Circles Touch Internally	$|C_{1}C_{2}|=|r_{1}-r_{2}|$
    Circles Intersect in Two Real Distinct Points	$|r_{1}-r_{2}|<|C_{1}C_{2}|<r_{1}+r_{2}$
    Circles do not Touch	(i) $|C_{1}C_{2}|>r_{1}+r_{2}$ (ii) $|C_{1}C_{2}|<|r_{1}-r_{2}|$

• Parabola Show/Hide Details
  A set of points in a plane such that the distance of each point from a fixed point (Focus, F) is equal to its distance from a fixed straight line (Directrix, L).

  • Terms Related to Parabola Show/Hide Details visualize
    • Axis of Parabola: The line through focus and perpendicular to directrix.
    • Vertex: The midpoint of the perpendicular line joining focus and directrix.
    • Focal Chord: A chord of the parabola through focus.
    • Latus Rectum: The focal chord perpendicular to the axis of the parabola.
    • Eccentricity: The ratio of the distance of any point on the parabola to its distance from directrix.
  • Main Facts about Standard Forms of Parabola Show/Hide Details

    Property	$y^{2}=4ax$	$y^{2}=-4ax$	$x^{2}=4ay$	$x^{2}=-4ay$
    Focus	$(a,0)$	$(-a,0)$	$(0,a)$	$(0,-a)$
    Vertex	$(0,0)$	$(0,0)$	$(0,0)$	$(0,0)$
    Length of Latus Rectum	$4a$	$4a$	$4a$	$4a$
    Equation of Latus Rectum	$x=a$	$x=-a$	$y=a$	$y=-a$
    Equation of Directrix	$x=-a$	$x=a$	$y=-a$	$y=a$
    Axis	$y=0$	$y=0$	$x=0$	$x=0$
    Eccentricity	1	1	1	1
    Graph	(Opens right)	(Opens left)	(Opens up)	(Opens down)

  • Parametric Form of Equations of Parabola Show/Hide Details

    Standard Form	Parametric Form
    $y^{2}=4ax$	$x=at^{2}$, $y=2at$
    $y^{2}=-4ax$	$x=-at^{2}$, $y=2at$
    $x^{2}=4ay$	$x=2at,$ $y=at^{2}$
    $x^{2}=-4ay$	$x=2at$, $y=-at^{2}$

  • Theorems on Parabola Show/Hide Details
    • The point on the parabola closest to the focus is the vertex.
    • The ordinate at any point P of the parabola is a mean proportional between the length of the latus rectum and the abscissa of P.
    • The tangent at any point P of a parabola makes equal angles with the line PF and the line through P parallel to the axis of parabola, F being focus.
  • Tangents and Normals (Parabola) Show/Hide Details
    • Equation of tangent to $y^{2}=4ax$: $yy_{1}=2a(x+x_{1})$.
    • Equation of normal to $y^{2}=4ax$: $y-y_{1}=-\frac{y_{1}}{2a}(x-x_{1})$.
  • Condition for $y=mx+c$ Tangent to Parabola Show/Hide Details

    Equation	Condition
    $y^{2}=4ax$	$c=\frac{a}{m}, m\ne0$
    $y^{2}=-4ax$	$c=-\frac{a}{m}, m\ne0$
    $x^{2}=4ay$	$c=-am^{2}$
    $x^{2}=-4ay$	$c=am^{2}$

    The line $y=mx+c$ will intersect parabola $y^{2}=4ax$ at:

    • Two distinct points if $c<\frac{a}{m}$.< /li>
    • A single point if $c=\frac{a}{m}$.
    • No point at all if $c>\frac{a}{m}$.

    The line $y=mx+c$ will be normal to parabola $y^{2}=4ax$ if $c=-2am-am^{3}$.

• Ellipse Show/Hide Details
  A set of points such that the distance of each point from a fixed point (Focus, F) bears a constant ratio (Eccentricity, 0 < e < 1) to its perpendicular distance from a fixed straight line (Directrix, L).

  • Terms Related To Standard Ellipse Show/Hide Details visualize
    • Vertices: Points on the standard ellipse where it crosses the major axis.
    • Covertices: Points on the standard ellipse where it crosses the minor axis.
    • Center: The midpoint of the line joining vertices (or covertices or Foci).
    • Major Axis: The line joining vertices.
    • Minor Axis: The line joining covertices.
    • Latera Recta: Chords perpendicular to major axis passing through foci.
    • Eccentricity: The ratio of the distance of any point on the ellipse from the focus to its distance from the directrix.
  • Main Facts about Standard Forms of Ellipse Show/Hide Details

    Property	$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b$	$\frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1, a>b$
    Foci	$(\pm c,0)$ ($c^{2}=a^{2}-b^{2}$)	$(0,\pm c)$ ($c^{2}=a^{2}-b^{2}$)
    Eccentricity	$0<e=\frac{c}{a}<1$	$0<e=\frac{c}{a}<1$
    Vertices	$(\pm a,0)$	$(0,\pm a)$
    Co vertices	$(0,\pm b)$	$(\pm b,0)$
    Center	$(0,0)$	$(0,0)$
    Length of Major Axis	$2a$	$2a$
    Equation of Major Axis	$y=0$	$x=0$
    Length of Minor Axis	$2b$	$2b$
    Equation of Minor Axis	$x=0$	$y=0$
    Length of Latus Rectum	$\frac{2b^{2}}{a}$	$\frac{2b^{2}}{a}$
    Equation of Directrices	$x=\pm\frac{c}{e^{2}}$	$y=\pm\frac{c}{e^{2}}$
    Graph	(Horizontal)	(Vertical)

  • Parametric Form of Equations of Ellipse Show/Hide Details

    Standard Form	Parametric Form
    $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$	$x=a~cos~\theta, y=b~sin~\theta$
    $\frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1$	$x=b~cos~\theta, y=a~sin~\theta$
    $\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$	$x=h+a~cos~\theta, y=k+b~sin~\theta$

  • Theorems (Ellipse) Show/Hide Details
    • The sum of the focal distances of any point on an ellipse is equal to length of major axis ($2a$).
    • The distance between centre and any focus of the ellipse is denoted by c, and is given as; $c=\sqrt{a^{2}-b^{2}}$.
    • The distance between foci is $2c$.
  • Tangents and Normals (Ellipse) Show/Hide Details
    • Equation of tangent to Ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$: $\frac{xx_{1}}{a^{2}}+\frac{yy_{1}}{b^{2}}=1$.
    • Equation of normal to Ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$: $\frac{a^{2}x}{x_{1}}-\frac{b^{2}y}{y_{1}}=a^{2}-b^{2}$.
  • Condition for $y=mx+c$ Tangent to Ellipse Show/Hide Details

    Equation	Condition
    $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$	$c^{2}=a^{2}m^{2}+b^{2}$
    $\frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1$	$c^{2}=a^{2}+m^{2}b^{2}$

    The line $y=mx+c$ will intersect Ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at:

    • Two distinct points if $c^{2}<a^{2}m^{2}+b^{2}$.
    • A single point if $c^{2}=a^{2}m^{2}+b^{2}$.
    • No point at all if $c^{2}>a^{2}m^{2}+b^{2}$.
• Hyperbola Show/Hide Details
  A set of points such that the distance of each point from a fixed point (focus, F) bears a constant ratio (Eccentricity, $e>1$) to its perpendicular distance from a fixed straight line (directrix, L).

  • Terms Related To Standard Hyperbola Show/Hide Details visualize
    • Vertices: Points on the standard hyperbola where it crosses the x-axis.
    • Center: Midpoint of the line joining vertices (or co vertices).
    • Transverse Axis: The line joining vertices.
    • Conjugate Axis: The line joining co vertices.
    • Latera Recta: Chords perpendicular to major axis passing through foci.
    • Eccentricity: Ratio of the distance of any point on the ellipse from the focus to its distance from the directrix.
    • Asymptotes: Lines that approach a curve but never touch; every hyperbola has two.
      • Slope of asymptotes: $\pm\frac{b}{a}$ (hyperbola along x-axis).
      • Slope of asymptotes: $\pm\frac{a}{b}$ (hyperbola along y-axis).
  • Main Facts about Standard Forms of Hyperbola Show/Hide Details

    Property	$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$	$\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$
    Foci	$(\pm c,0)$ ($c^{2}=a^{2}+b^{2}$)	$(0,\pm c)$ ($c^{2}=a^{2}+b^{2}$)
    Eccentricity	$e=\frac{c}{a}>1$	$e=\frac{c}{a}>1$
    Vertices	$(\pm a,0)$	$(0,\pm a)$
    Center	$(0,0)$	$(0,0)$
    Length of Transverse Axis	$2a$	$2a$
    Equation of Transverse Axis	$y=0$	$x=0$
    Length of Conjugate Axis	$2b$	$2b$
    Equation of Conjugate Axis	$x=0$	$y=0$
    Length of Latus Rectum	$\frac{2b^{2}}{a}$	$\frac{2b^{2}}{a}$
    Equation of Directrices	$x=\pm\frac{c}{e^{2}}$	$y=\pm\frac{c}{e^{2}}$
    Equation of Asymptotes	$y=\pm\frac{b}{a}x$	$y=\pm\frac{a}{b}x$

  • Rectangular Hyperbola Show/Hide Details
    A hyperbola is said to be rectangular if its conjugate and transverse axis are equal ($2a=2b \Rightarrow a=b$).

    • Equation: $x^{2}-y^{2}=a^{2}$.
    • Asymptotes are perpendicular to each other, equation $y=\pm x$.
    • Eccentricity: $e=\sqrt{2}$.
    • Equation referred to its asymptotes as co-ordinate axis: $xy=c^{2}$ where $c^{2}=\frac{a^{2}}{2}$.
    • Parametric equation: $x=ct, y=\frac{c}{t}$.
  • Parametric Form of Equations of Hyperbola Show/Hide Details

    Standard Form	Parametric Form
    $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$	$x=a~sec~\theta, y=b~tan~\theta$
    $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$	$x=b~tan~\theta, y=a~sec~\theta$
    $\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1$	$x=h+a~sec~\theta, y=k+b~tan~\theta$

  • Theorems (Hyperbola) Show/Hide Details
    • The difference of the focal distances of any point on the hyperbola is equal to the length of transverse axis ($2a$).
    • The distance between centre and any focus of the hyperbola is $c=\sqrt{a^{2}+b^{2}}$.
    • The distance between the foci is $2c$.
    • Angle between the asymptotes is $2 \tan^{-1}\left(\frac{b}{a}\right)$.
    • The product of distances from the foci to any tangent to the hyperbola is $b^2$.
  • Tangents and Normals (Hyperbola) Show/Hide Details
    • Equation of tangent to $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$: $\frac{xx_{1}}{a^{2}}-\frac{yy_{1}}{b^{2}}=1$.
    • Equation of normal to $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$: $\frac{a^{2}x}{x_{1}}+\frac{b^{2}y}{y_{1}}=a^{2}+b^{2}$.
  • Condition for $y=mx+c$ Tangent to Hyperbola Show/Hide Details

    Equation	Condition
    $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$	$c^{2}=a^{2}m^{2}-b^{2}$
    $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$	$c^{2}=a^{2}-m^{2}b^{2}$

• Forms of Conics from General Equation Show/Hide Details

  The general second-degree form: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] represents different conic sections depending on the values of \( A \), \( B \), and \( C \). Below are the simplified cases of conics in the form \( A(x-h)^2 \pm B(y-k)^2 = C \):

  Equation Form	Conditions	Conic Type
  \( A(x - h)^2 - B(y - k)^2 = C \)	\( A, B > 0;\ C > 0 \)	Hyperbola (horizontal)
  \( B(y - k)^2 - A(x - h)^2 = C \)	\( A, B > 0;\ C > 0 \)	Hyperbola (vertical)
  \( A(x - h)^2 + B(y - k)^2 = C \)	\( A, B > 0;\ C > 0 \)	Ellipse
  \( A(x - h)^2 + A(y - k)^2 = C \)	\( A > 0;\ C > 0 \)	Circle
  \( A(x - h)^2 - B(y - k)^2 = 0 \)	\( A, B > 0;\ C = 0 \)	Degenerate Hyperbola (pair of lines)
  \( A(x - h)^2 + B(y - k)^2 = 0 \)	\( A, B > 0;\ C = 0 \)	Degenerate Ellipse (point)
  Only one squared term	\( A \ne 0,\ B = 0 \) or vice versa	Parabola
  No squared terms	\( A = B = 0 \)	Straight Line

  Key Rule: A conic is a hyperbola if it includes two squared terms with opposite signs. It's an ellipse or circle if both squared terms are positive and coefficients match (circle).

• Foci of Conic Sections (Shifted Center) Show/Hide Details

  Given a conic centered or vertexed at \( (h, k) \), here are the formulas for the foci positions:

  Conic	Standard Equation	Foci	Where
  Circle	\( (x - h)^2 + (y - k)^2 = r^2 \)	Only center: \( (h, k) \)	No separate foci — every point equidistant from center
  Ellipse (horizontal)	\( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1,\ a > b \)	\( (h \pm c,\, k) \), where \( c = \sqrt{a^2 - b^2} \)	Along major axis (horizontal)
  Ellipse (vertical)	\( \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1,\ a > b \)	\( (h,\, k \pm c) \), where \( c = \sqrt{a^2 - b^2} \)	Along major axis (vertical)
  Hyperbola (horizontal)	\( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \)	\( (h \pm c,\, k) \), where \( c = \sqrt{a^2 + b^2} \)	Along transverse axis (horizontal)
  Hyperbola (vertical)	\( \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \)	\( (h,\, k \pm c) \), where \( c = \sqrt{a^2 + b^2} \)	Along transverse axis (vertical)
  Parabola (horizontal)	\( (y - k)^2 = 4a(x - h) \)	\( (h + a,\, k) \)	Focus lies along x-axis direction
  Parabola (vertical)	\( (x - h)^2 = 4a(y - k) \)	\( (h,\, k + a) \)	Focus lies along y-axis direction

  Note: \( c \) represents distance from center to focus. Always check if \( a > b \) for ellipses to determine major axis direction.

• Condition for y = mx + c tangent to conic Show/Hide Details
  • General Show/Hide Details

    To check whether the line \( y = mx + c \) is tangent to any conic section:

    1. Substitute \( y = mx + c \) into the conic’s equation.
    2. Simplify to get a quadratic equation in \( x \).
    3. Compute the discriminant \( D = b^2 - 4ac \).

    • \( D = 0 \): Line is tangent to the conic.
    • \( D > 0 \): Line intersects conic at two points (secant).
    • \( D < 0 \): Line does not intersect conic.

    This method works for any conic including general forms like:

    \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]

  • Slope Form of Tangents for Conics Show/Hide Details

    The following are standard equations of tangent lines with slope \(m\) for different conics:

    Conic	Equation
    Circle: \(x^2 + y^2 = r^2\)	\(y = mx \pm \sqrt{r^2(1 + m^2)}\)
    Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)	\(y = mx \pm \sqrt{a^2m^2 + b^2}\)
    Hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)	\(y = mx \pm \sqrt{a^2m^2 - b^2}\)
    Hyperbola: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)	\(y = mx \pm \sqrt{a^2 - m^2b^2}\)

    Each form comes from substituting \(y = mx + c\) into the conic and applying the condition \(D = 0\) for tangency.

  • Circle Show/Hide Details
    The line $y=mx+c$ touches the circle $x^{2}+y^{2}=a^{2}$ if $c=\pm a\sqrt{1+m^{2}}$.

    • Intersects in two distinct points if $a^{2}(1+m^{2})-c^{2}>0$.
    • Intersects in real and coincident points if $a^{2}(1+m^{2})-c^{2}=0$.
    • Intersects in imaginary points if $a^{2}(1+m^{2})-c^{2}<0$.< /li>
  • Ellipse Show/Hide Details

    Equation	Condition
    $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$	$c^{2}=a^{2}m^{2}+b^{2}$
    $\frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1$	$c^{2}=a^{2}+m^{2}b^{2}$

    The line $y=mx+c$ will intersect Ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at:

    • Two distinct points if $c^{2}<a^{2}m^{2}+b^{2}$.
    • A single point if $c^{2}=a^{2}m^{2}+b^{2}$.
    • No point at all if $c^{2}>a^{2}m^{2}+b^{2}$.
  • Parabola Show/Hide Details

    Equation	Condition
    $y^{2}=4ax$	$c=\frac{a}{m}, m\ne0$
    $y^{2}=-4ax$	$c=-\frac{a}{m}, m\ne0$
    $x^{2}=4ay$	$c=-am^{2}$
    $x^{2}=-4ay$	$c=am^{2}$

    The line $y=mx+c$ will intersect parabola $y^{2}=4ax$ at:

    • Two distinct points if $c<\frac{a}{m}$.
    • A single point if $c=\frac{a}{m}$.
    • No point at all if $c>\frac{a}{m}$.

    The line $y=mx+c$ will be normal to parabola $y^{2}=4ax$ if $c=-2am-am^{3}$.

  • Hyperbola Show/Hide Details

    Equation	Condition
    $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$	$c^{2}=a^{2}m^{2}-b^{2}$
    $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$	$c^{2}=a^{2}-m^{2}b^{2}$

• Equation of Tangent to a Curve Show/Hide Details
  Generally, to find an equation of tangent at a point $P(x_{1},y_{1})$, make the replacements in the given equation of the curve:

  • $x^2$ by $xx_1$.
  • $y^2$ by $yy_1$.
  • $xy$ by $\frac{xy_1+yx_1}{2}$.
  • $x$ by $\frac{x+x_1}{2}$.
  • $y$ by $\frac{y+y_1}{2}$.
• Identification of a Conic From the General Equation Show/Hide Details
  The general equation of a conic $ax^2+2hxy+by^2+2gx+2fy+c=0$ represents:

  • A pair of straight lines, if $\Delta = 0$.
  • An ellipse or circle if $h^2-ab<0, \Delta\ne 0$.
  • A parabola if $h^2-ab=0, \Delta\ne 0$.
  • A hyperbola if $h^2-ab>0, \Delta\ne 0$.
  • Centre of general conic is $\left(\frac{hf-bg}{ab-h^2}, \frac{gh-af}{ab-h^2}\right)$.
• LOCUS Show/Hide Details
  A locus is a set of points satisfying given condition(s).

  • 2-DIMENSIONAL
    • Set of all points in a plane at a constant distance from a fixed point: circle.
    • Set of all points in a plane at a constant distance from a given straight line: a pair of straight lines parallel to the given line.
    • Set of all points in a plane such that the sum of their distances from two fixed points is a constant: an ellipse.
    • Set of all points in a plane such that the difference of their distances from two fixed points is a constant: a hyperbola.
    • Set of all points such that their distances from a fixed point are equal to their distances from a fixed straight line: a parabola.
    • Locus of a point in a plane that is equidistant from two fixed points: the perpendicular bisector (mediator) of the straight line joining those two points.
    • Locus of a point in a plane equidistant from two intersecting straight lines: pair of straight lines bisects the angles between two given straight lines.
  • 3-DIMENSIONAL
    • Set of all points in space at a constant distance from a fixed point: a sphere.
    • Set of all points in space at a constant distance from a given straight line: a cylinder.
    • In space an ellipse is changed into an ellipsoid.
    • In space hyperbola is shifted to a hyperboloid.
    • In space parabola is called paraboloid.
• Transformations Show/Hide Details
  • Translation of Axis: To transform the given equation referred to the new origin $O'(h,k)$, put $x=X+h, y=Y+k$ in the given equation of the curve.
  • Rotation of Axis: To find the new equation when axis are rotated through an angle $\theta$, put $x=X~cos~\theta-Y~sin~\theta, y=X~sin~\theta+Y~cos~\theta$ in the given equation.
  • Elimination of the Xy-Term: To remove the xy-term from a second degree equation, determine the value of $\theta$ such that $\tan{2\theta}=\frac{2h}{a-b}$.