Slide 2: Objective:- To study about coordinate geometry.
1. The straight line
2. The circle
4. Hyperbola THE NAME COORDINATE GEOMETRY : THE NAME COORDINATE GEOMETRY The equation of a curve represents the fundamental properties of different points on the curve which we locate by means of real no, called coordinates and so, instead of giving this new type of geometry the name analytical geometry, we can call it coordinate geometry or plane cartesian geometry, the former name being more commonly used ,and the latter is after the name of the inventor Descartes. DEFINATION OF COORDINATE GEOMETRY : DEFINATION OF COORDINATE GEOMETRY It is that branch of geometry in which two numbers , called coordinates, are used to indicate the position of a point in a plane and which makes use of algebraic methods in the study of geometric figures. Rectangular Axes : Rectangular Axes The position of a point in a plane is fixed by selecting the axes of reference which are formed by combining the two number scales at right angle to each other so that their zero point coincide . The horizontal number scale is called the x-axis & the vertical number scale the y-axis .The point where two scales cross each other is called the origin. The two together are called rectangular axes. Coordinates Definition : Coordinates Definition The position of each point of the plane is determined with reference to the rectangular axes by means of a pair of numbers called co-ordinates which are distances of the from the axes. X Y
The distance of the point from the y-axis is called the x-coordinate or abscissa & the distance of the point from x-axis is called y-coordinate or ordinate. N P O M Y X QUADRANTS : QUADRANTS The coordinate axes separate the plane into four regions called quadrants. By custom the quadrants are numbered I, II , III & IV in the counter-clock wise direction, as shown in figure. Y Y’ X X’ I II III I\/ CONVENTION OF SIGNS : CONVENTION OF SIGNS A point of the plane may lie in any quadrant and falling back to our number scales, it is easily understood that-
(i)For distance along the x-axis, positive values are measured to the right of the origin, and negative values to its left;
(ii)For distances along y-axis, positive values are measured upward and negative values downward. PLOTTING A POINT : PLOTTING A POINT The points can be plotted by measuring its proper distances from the axes. Thus any point (h , k) can be plotted as follows:
1.Measure OM equal to h along x-axis
2.Now measure MP perpendicular to OM & equal to k
Observe the rules of signs in both cases. DISTANCE FORMULA : DISTANCE FORMULA The distance d between the points p(x , y) & Q(x’ , y’) is given by the formula
d = v(x2-x1)2 + (y2-y1 )2
Let P(x1,y1) and Q(x2,y2) be the two points. From P,Q draw PL,QM perpendiculars on the x-axis and PR perpendicular on MQ.
From rt.PQR, PQ2 = PR2+RQ2
Or, d2=v(X2-X1)2+(Y2-Y1)2 COMPLETE DISTANCE FOMULA : COMPLETE DISTANCE FOMULA D2=(x2-x1)2+(y2-y1)2 : d=+v (x2-x1)2+(y2-y1)2
We take only the positive square root as we are usually interested only in the magnitude to the segment PQ.
We have derived this formula taking P and Q in the first quadrant but the formula is true in whatever quadrants the points may lie. In coordinate geometry all the theorems and formula proved by supposing points or lines to be in the first quadrant will always remain true in whatever quadrants the points or lines may lie. Only the proper sign will have to taken with the coordinates. DIVISION or SECTION FORMULAE : DIVISION or SECTION FORMULAE To find the coordinates of the point which divides internally the line joining two given points in a given ratio.
Let A(x1,y1) and B(x2,y2) be the given points and P a point on AB which divides it in the given ratio m1 : m2. It is required to find the co-ordinates of P. Suppose they are(x,y) .Draw the perpendiculars AL,PM,BN on OX, and AK,PT on PM and BN respectively. Then, from similar triangles APK and PBT, we have
AP/PB = AK/PT = KP/TB THE STRAIGHT LINE : THE STRAIGHT LINE DEF:- A straight line is a curve such that every point on the line segment joining any two points on it lies on it.
ax+by+c=0 represent a straight line. SLOPE (GRADIENT )OF A LINE
The trigonometrical tangent of the angle that a line makes with the positive direction of the x-axis in anticlockwise sense is called the slope or gradient of the line. m= tan ?
SLOPE OF LINE IN TERMS OF COORDINATES OF ANY TWO POINTS ON IT.
m=y2-y1 / x2-x1 EQUATION OF LINE PARALLEL TO THE AXIS:
The equation of line parallel to x-axis at a distance b from it is y=b
The equation of line parallel to y-axis at a distance a from it is x=a Y=b X=a y x y x DIFFERENT FORMS OF THE EQUATION OF A STRAIGHT LINE : DIFFERENT FORMS OF THE EQUATION OF A STRAIGHT LINE THE SLOPE INTERCEPT FORM OF A LINE:-
The equation of a line with slope m and making an intercept c on y –axis is y =mx + c THE POINT SLOPE FORM OF A LINE :-
The equation of a line which passes through the point (x1,y1) and has the slope m is y-y1=m(x-x1) THE TWO-POINT FORM OF A LINE :-
The equation of a line which passes through the point(x1,y1) and (X2,Y2) is given by
y-y1=y2-y1 / x2-x1 (x-x1) Slope m (0.c) x y x y P’(x’, y’) P(x,y) P(x2,y2) P’(x’,y’) P’’(x1,y1) Slide 15: THE NORMAL FORM OR PERPENDICULAR FORM OF A LINE:-
The equation of the straight line upon which length of the perpendicular from the origin is p and this perpendicular makes an angle ? with x-axis is x cos ? +y sin ? =p THE INTERCEPT FORM OF A LINE :-
The equation of a line which cuts off intercepts a and b respectively from the x and y-axes is x/a + y/b=1 DISTANCE OF A POINT FROM A LINE:-
d=| (Ax +By + c)|/ v(A2+B2) DISTANCE BEWEEN TWO PARALLEL LINES:-
d=| c1-c2|/ v(1+m2)
d= | c1-c2|/ v(A2+B2) (0,b) (a,0) (0,0) p ? y x y x P (x1,y1) R (0,-C/B) Q(-C/A,0) L: Ax+By+c=0 Y=mx+c1 Y=mx+c2 p The circle : The circle A circle is defined as the locus of a point which moves in a plane such that its distance from a fixed point in that plane is always constant. STANDARD EQUATION OF THE CIRCLE
Central equation of the circle:-
2. Centre at origin
x2+y2=r2 GENERAL EQUATION OF THE CIRCLE
The equation x2+y2+2gx+2fy+c=0 always represent a circle whose centre is (-g,-f) and radius = v (g2+f2-c ) INTCEPTS ON THE AXES
The lengths of intercepts made by the circle x2+y2+2gx+2fy+c=0 with X and Y axes are
2 v (g2-c) and 2v(f2-c) respectively (h , k) parabola : parabola CONIC SECTION:- A conic section of conic is the locus of a point P which moves in such a way that its distances from a fixed point S always bears a constant ratio to its distance from a fixed line, all being in the same plane. VARIOUS TERMS OF CONIC SECTION:-
FOCUS: The fixed points is called the focus of the conic section
DIRECTRIX: The fixed straight line is called the directrix of the conic section.
ECCENTRICITY : The constant ratio is called the eccentricity of the conic section and is denoted by e.
AXIS: The straight line passing through the focus and perpendicular to the directrix is called axis of the conic section.
VERTEX: The points of intersection of the conic section and the axis are called vertices of the conic section.
CENTRE: T he point which bisects every chord of the conic passing it , is called the centre of the conic section.
LATUS-RECTUM: The latus-rectum of a conic is the chord passing through the focus and perpendicular to the axis. Slide 18: As mentioned above the eccentricity of a conic is graphically represented by e
1.For e<1 , the conic obtained is an ellipse
2. For e=1 , the conic obtained is a parabola
3. For e>1 , the conic obtained is a hyperbola
4. For e=0 , the conic obtained is a circle. Slide 19: THE PARABOLA:- A parabola is the locus of a point of a point which moves in a plane such that its distance from a fixed point in the plane is plane is always equal to its distance from a fixed straight line in the same plane. EQUATION OF THE PARABOLA IN PARABOLA IN ITS STANDART FORM
*? y2=4ax (a>0) DOUBLE ORDINATE:- A chord passing through P perpendicular to the axis of the parabola is called the double ordinate through point P
LATUSRECTUM:-A double ordinate through the focus is called the latusrectum i.e.., the latusrectum of a parabola is a chord passing through the focus perpendicular to the axis FOCUS DISTANCE OF ANY POINT: The distance of P(x , y) from the focus S is called the focal distance of the point P.
*? (a + x) is the distance of the point P(x, y) Slide 20: SOME OTHER STANDARD FORM OF PARABO LA Ellipse : Ellipse ELLIPSE: - An ellipse is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus ) in the same plane to its distance from a fixed straight line (called directrix) is always constant which is always less than unity. *? EQUATION OF THE ELLIPSE IN THE STANDARD FORM x2/a2 + y2/b2=1 x x’ y y’ O F F 1 2 Vertices , major and minor axes , foci ,directories and centre of the ellipse x2/a2 + y2/b2=1 ( a>b) : Vertices , major and minor axes , foci ,directories and centre of the ellipse x2/a2 + y2/b2=1 ( a>b) VERTICES:- The points A and A’ where the curve meets the line joining the foci S and S’ are called the ellipse. The coordinates of A and A’ are(a,0) and (-a.0) respectively
MAJOR AND MINOR AXIS:- The distance AA’=2a and BB’=2a are called the major and minor axis of the ellipse.
FOCI:- The points S (ae,0) and S’(-ae,0) are the foci of the ellipse.
DIRECTRICES:- ZK and Z’K’ are two directories of the ellipse and their equation are x=a/c and x= -a/c respectively.
CENTRE:- The point C(0,0) where two axis intersects.
EECENTRICITY OF THE ELLIPSE x2/a2 + y2/b2=1 ( a>b) Fr the ellipse x2/a2 + y2/b2=1 ( a>b)
e = v1-(major axis/ minor axis)2
LATUSRECTUM : The double ordinate passing through the focus
FOCAL DISTANCES OF A POINT ON THE ELLIPSE:
The sum of the focal distances of any point on an ellipse is constant and equal to the length of the major axis of the ellipse. i.e.., 2a=constant Equation of the ellipse in the other forms : Equation of the ellipse in the other forms x2/b2 + y2/a2=1 The relation between semi major axis and semi minor axis and the distance of the focus from the centre of the ellipse.
b2=a2(1-e2) X X’ Y’ Y O F 2 F 1 HYPERBOLA : HYPERBOLA DEF:- The hyperbola the set of all points in a plane, the differences of whose distances from two fix points in the plane is constant STANDARD EQUATION OF THE HYPERBOLA:
x2/a2-y2/b2=1 (when the foci are along x- axis.)
y2/a2-x2/b2=1 (when the foci are along y-axis.) O O Y Y Y’ Y’ X’ X’ X X Some important point about hyperbola : Some important point about hyperbola A hyperbola in which a=b is a called an equilateral hyperbola.
The standard equations of hyperbolas have transverse and conjugate axes as the coordinate axes and the centre at the origin. LATUS RECTUM:- Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola. ECCENTRICITY OF THE HYPERBOLA:-
e= v 1+(conjugate axis/transverse axis)2 Slide 27: Presented By:-
Devi Prasad Barick