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Conic Section

Cone & its Sections

• Right Circular Cone: The surface generated when one straight line that intersects another fixed straight line is rotated at an oblique angle. The lines are called generators of the cone.
• Axis of the Cone: The fixed line of the cone is called its axis.
• Elements of the Cone: The possible positions of the generating line in its rotation about the axis are its elements.
• Vertex: The common intersection point of all the cone's elements is called Vertex of cone.
• Napes: The two symmetrical parts of the generated surface on each side of the vertex are cone's Napes.
• Conic Sections or Conics: The curves that can be obtained by cutting a cone with a plane are called conic sections or conics.
• Circle: When the intersecting plane cuts completely across one napes at right angle to the axis of cone, the curve is called circle.
• Point Circle: If the intersection plane cuts across the vertex of a cone, the resulting curve is called point circle.
• Parabola: When the intersecting plane is parallel to an element, an passing through only one nape, the resulting curve is Parabola.
• Ellipse: When the intersecting plane cuts completely across one nape at an oblique angle to the axis, the curve is an ellipse.
• Hyperbola: When the intersecting plane cuts through both napes parallel to the axis of the cone the resulting curve is hyperbola.

Circle

A set of points such that distance of each point fixed point called center remains constant. The constant distance is called radius of the circle.

Equation

• Standard Equation
  When center is \((h,k)\) & radius is \(r\): $$ (x-h)^2 + (y-k)^{2 = r}2 $$
  When center is \((0,0)\) & radius is \(r\): $$ x^2 + y^{2 = r}2 $$
• Parametric Equation \[ x= r\cos\theta \] \[ y= r\sin\theta \]
• If end points of the diameter of circle are \(a(x_,y_1)\), \(B(x_2,y_2)\) $$ (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0 $$
• General Form $$ x^2 + y^2 + 2gx+2fy+c=0 $$ The general form of circle equation have following properties:
  • It involves three constants \(f,g,c\)
  • It is a second degree equation in which coefficients of \(x^2\) & \(y^2\) are \(1\)
  • It does not involves \(xy\) product
  • Center of the circle is \((-g,-f)\)
  • Radius of circle is \(\sqrt{g^2+f^2-c}\). If:
    • \(r=\sqrt{g^2+f^2-c}>0\) circle is real circle
    • \(r=\sqrt{g^2+f^2-c}=0\) circle is point circle
    • \(r=\sqrt{g^2+f^2-c}<0\) circle is imaginary circle

Properties

• Perpendicular dropped from the center of a circle on a chord bisects the cord.
• 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 chord is perpendicular to the chord.
• Congruent chords of a circle are equidistant from its center.
• Measure of the central angle of a minor arc is double the measure of the angle subtended in the corresponding major arc.
• An angle is 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 center of the circle.
• Mid point of the hypotenuse of a right triangle is the circum-center of the triangle.
• Perpendicular dropped from a point of a circle on diameter is a mean proportional between the segments into which it divides the diameter.

Theorems

• the point \(P(x_1,y_2)\), on circle \(x^2+y^2+2gx+2fy+c=0\) lies:
  • outside circle if \(x_1^2+y_1^2+2gx+2fy+c>0\)
  • on circle if \(x_1^2+y_1^2+2gx+2fy+c=0\)
  • inside circle if \(x_1^2+y_1^2+2gx+2fy+c<0\)
• Any two tangents can be drawn to a circle from any point outside the circle.
• Let \(P(x_1,y_1)\) be a point outside the circle \(x^2+y^2+2gx+2fy+c=0\), then length of either tangents drawn to circle is : $$ \sqrt{x_1^2 +y_1^2 +2gx_1 +2fy_1 + c} $$
• The line \(y=mx+c\) intersects the circle \(x^2+y^2=a^2\) in at the most two points, the points are:
  • Real & distinct if \(a^2(1+m^2)>c^2\)
  • Real & coincident if \(a^2(1+m^2)=c^2\) (also called condition of tangency)
  • Imaginary if \(a^2(1+m^2)<c^2\)

Contact of two Circles

Let \(r_1\), \(r_2\) be the radii of two circles with \(C_1\) & \(C_2\) be their centers, then:

• Circles touches externally if \(|C_1C_2|=r_1+r_2\)
• Circles touches internally if \(|C_1C_2|=|r_1-r_2|\)
• Circles intersect in two real distinct points if \(|r_1-r_1|<|C_1C_2|<(r_1+r_2)\)
• Circle don't touches each other if \(|r_1-r_2|>|C_1C_2|>r_1+r_2\)

Tangent

A straight line that touches the curve at a single point without cutting the curve is called tangent. It is perpendicular to the line segment joining center to that point.

Equation

• Equation of tangent to the circle \(x^2+y^2=a^2\) at \(P(x_1,y_1)\): $$ xx_1+ yy_1=a^2 $$
• Equation of tangent to the circle \(x^2+y^2 +2gx+2fy+c=0\) at \(P(x_1,y_1)\): $$ xx_1+ yy_1 + g(x+x_1)+ f(y+y_1)+c=0 $$

Normal

A straight line perpendicular to the curve at point of tangency.

Equation

• Equation of normal to the circle \(x^2+y^2=a^2\) at \(P(x_1,y_1)\): $$ xx_1- yy_1=0 $$
• Equation of tangent to the circle \(x^2+y^2 +2gx+2fy+c=0\) at \(P(x_1,y_1)\): $$ (x-x_1)(y_1+f) + (y-y_1)(x_1+g)=0 $$

Parabola

A set of points in a plane such that the distance of each point form a fixed point (focus), equal to its distance from a fixed straight line (directrix, L)

Equation

• Standard Form of standard parabola: $$ y^2=4ax $$
• Parametric Equation: $$ x=at^2 $$ $$ y=2at $$
• General form: $$ ax^2 +by^2 +2gx+2fy+c=0 $$ Its a second degree equation in which either \(a=0\) or \(b=0\) but not both.

Terms

• Axis of Parabola: The line through focus and perpendicular to directrix is called axis of parabola.
• Vertex: The mid point of the perpendicular line joining focus and directrix is called vertex of parabola.
• Focal Chord: A chord of the parabola through focus is called focal chord.
• Latusrectum: The focal chord perpendicular to the axis of the parabola is called latusrectum of the parabola.
• Eccentricity: The ratio of the distance of any point on the parabola to its distance from directrix.

Summary of Standard Parabolas

#	1	2	3	4
Equation	\(y^2=4ax\)	\(y^2=-4ax\)	\(x^2=4ay\)	\(x^2=-4ay\)
Focus	\((a,0)\)	\((-a,0)\)	\((0,a)\)	\((0,-a)\)
Vertex	\((0,0)\)	\((0,0)\)	\((0,0)\)	\((0,0)\)
Length of Latusrectum	\(4a\)	\(4a\)	\(4a\)	\(4a\)
Equation of Latusrectum	\(x=a\)	\(x=-a\)	\(y=a\)	\(y=-a\)
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

Theorems on Parabola

• 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 latusrectum 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.

Ellipse

A set of point such that the distance of each from fixed point (focus, \(F\)) bears a constant ration (Eccentricity, \(0<e<1\)) to its perpendicular distance from a fixed straight line (directrix, \(L\)).

Terms

• Vertices: The points on the standard ellipse where it crosses the x-aix.
• Co-vertices: The points on the standard ellipse where it crosses the y-axis.
• Center: The midpoint of the line joining vertices
• Major axis: The lines joining vertices is called major axis.
• Minor axis: The line joining co-vertices is called minor axis.
• Latusrectum: The cords perpendicular to major axis passed through foci are called latusrectum.
• Eccentricity: The ratio of the distance of any point on the ellipse from the focus to its distance from the directrix.

Summary of Standard Forms of Ellipse

#	1	2
Equation	$$ \frac{x^2 }{a^2 } + \frac{y^2 }{b^2 }=1,\;a>b $$	$$ \frac{y^2 }{a^2 } + \frac{x^2 }{b^2 }=1,\;a>b $$
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 Major Axis	\(2b\)	\(2b\)
Equation of Major Axis	\(x=0\)	\(y=0\)
Length of Latusrectum	\({2b^2}/{a}\)	\({2b^2}/{a}\)
Equation of Directrix	\(x=\pm{c}/{e^2}\)	\(y\pm{c}/{e^2}\)
Graph

Theorems on Ellipse

• The sum of the focal distances of any point on an ellipse is equal to length of major axis \(2a\)
• The distance between center and any focus of the ellipse is denoted by \(c\), and is given as \(\sqrt{a^b-b^2}\)
• The distance between foci is \(2c\)

Hyperbola

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

• Vertices: The points on the standard hyperbola where it crosses the x-axis.
• Center: The midpoint of the line joining the vertices or co-vertices.
• Transverse axis: The line joining vertices is called transverse axis.
• Conjugate axis: The line joining co vertices is called conjugate axis.
• Latusrecta: The chords perpendicular to major axis passes through foci are called Latusrecta.
• Eccentricity: The ratio of distance of any point on ellipse from the focus to its distance from directrix,
• Asymptotes: In general, an asymptote is a line approaches to a curve but never touches it.
  Every hyperbola has associated with two lines called asymptotes. Their point of intersection is center of hyperbola.
  • Slope of asymptotes \(=\pm b/a\) (hyperbola opens to sides)
  • Slope of asymptotes \(=\pm a/b\) (hyperbola opens to up & down)

Summary of Standard Forms of Hyperbola

#	1	2
Equation	$$ \frac{x^2 }{a^2 }-\frac{y^2 }{b^2 }=1 $$	$$ \frac{y^2 }{a^2 }-\frac{x^2 }{b^2 }=1 $$
Foci	\((\pm c,0),c^2=a^2+b^2\)	\((0, \pm c),c^2=a^2+b^2\)
Eccentricity	\(e=c/a>1\)	\(e=c/a>1\)
Vertices	\((\pm a,0)\)	\((0,\pm a)\)

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