Section of a Cone
* If a straight line intersects a fixed vertical line at a fixed point and rotates such that the angles between the two lines remains constant, then the resulting surface is called a double right circular cone.
* The parts of a cone include a vertical axis, a generator line, a vertex and an upper nappe and a lower nappe.
* When a plane intersects a cone, it cuts a section from the cone. This section is
called a conic section or a conic.
* Depending on a position of the intersecting plane and the angle it makes with the
axis, a conic section can be a circle, an ellipse, a parabola or a hyperbola.
Circle
* The equation of a circle is given by (x – h)2 + (y – k)2 = r2,
Where:
* (h, k) are the coordinates of the centre of the circle
* R is the radius of the circle
The equation of a circle whose centre lies at the origin (0, 0) is given by x^{2} + y^{2} = r^{2} ,
where:
* r is radius of the circle
The general form of the equation of a circle is:
X2 + y2 + 2gx + 2ƒy + c =0, with centre (-g, -ƒ)
And radius \sqrt{g² + f² - c}.
An equation of the form x2+ y2+ 2gx + 2ƒy + c = 0
Represent a circle only if g2 + ƒ2 – c ≥ 0.
If g2 + ƒ2 – c > 0, then the equation
X2+ y2 + 2gx + 2ƒy + c = 0 represents a real circle with centre (-g, -ƒ) and radius
\sqrt{g² + f² - c}
If g2 + ƒ2 – c = 0, then the equation
X2 + y2 +2gx + 2ƒy + c = 0 represents a point circle.
If g2 + ƒ2 – c < 0, then the equation
X2+ y2+ 2gx + 2ƒy + c = 0 represents an imaginary circle with a real centre.
Parabola and Its Standard Equations
* A parabola is a set of all the points in a plane, which are equidistant from a fixed line and a fixed point.
Parts of a parabola:
* Focus
* Directrix
* Axis of symmetry
* Vertex
* Latus Rectum
Standard Equation of a parabola:
* y2= 4ax
* y2= – 4ax
* x2= 4ay
*x2= -4ay
Latus Rectum = 4a where:
a = Distance between vertex and focus
Ellipse
An ellipse is the set of all those points in a plane the sum of whose distance from two fixed points is constant.
An ellipse has two foci, a major axis, a centre and two vertices.
Semi-major axis = \frac{1}{2} major axis
Semi-minor axis = \frac{1}{2} minor axis
In an ellipse, c = \sqrt{a² - b² }, where:
* a = Semi-major axis
* b = Semi-minor axis
* c = distance of focus of am ellipse from centre
If c = 0, then a = b, and the ellipse becomes a circle.
If c = a, then b = 0, and the ellipse becomes a straight line.
Standard Equation of an Ellipse
P_{1} F_{1} + P_{1} F_{2} = P_{2} F_{1} + P_{2} F_{2} = P_{3} F_{1} + P_{3} F_{2} = 2a , Where
* P_{1} P_{2} P_{3} any three points on the ellipse
* F_{1} and F_{2} are foci of the ellipse and
* 2a is length of major axis of the ellipse.
Eccentricity of an ellipse (e) = \frac{c}{a}
* The standard equation of an ellipse with centre (0, 0) and major axis along the X-axis is:
\frac{x²}{a²} + \frac{y²}{b²} = 1
* The standard equation of an ellipse with centre (0, 0) and major axis along the Y-axis is:
\frac{x²}{b²} + \frac{y²}{a²} = 1
Length of latus rectum of an ellipse = \frac{2b²}{a} , where
a = Length of semi-major axis
b = Length of semi-minor axis
Hyperbola
* A hyperbola is set of all the points in a plane the difference of whose distances from two fixed points in the plane is constant.
* A hyperbola has two foci, a transverse axis, a conjugate axis, a centre and two vertices.
* The distance between two vertices of a hyperbola is considered the length of its transverse axis.
* In a hyperbola, b = \sqrt{c² - a²} , where
b= Semi-conjugate axis
c = Distance of a focus of a hyperbola from its centre
a = Semi-transverse axis
PF_{1} - PF_{2} = 2a, where:
* P is any point on the hyperbola
*F_{1} and F_{2} are the foci of hyperbola and
* 2a is the length of transverse axis of hyperbola
Standard Equations of Hyperbola
P_{1} F_{2} – P_{1} F_{1} = P_{2} F_{2} – P_{2} F_{1} = P_{3} F_{1} – P_{3} F_{2} = 2a
Eccentricity of a hyperbola (e) = \frac{c}{a} , where
c = Distance of a focus from the centre
a = Length of semi-transverse axis
Standard equation of hyperbola with centre (0,0) and transverse axis along the X-axis is:
\frac{x²}{a²} - \frac{y²}{b²} = 1
Standard equation of a hyperbola with centre (0, 0) and transverse axis along the
Y-axis is:
\frac{y²}{a²} - \frac{x²}{b²} = 1
Length of lactus rectum of a hyperbola = \frac{2b²}{a} , where
a = Length of semi-transverse axis
b = Length of semi-conjugate axis