ncert books for class maths cbsc

A Note The standard equations of parabolas have focus on one of the coordinate axis; vertex at the origin and thereby the directrix is parallel to the other coordinate axis. However, the study of the equations of parabolas with focus at any point and any line as directrix is beyond the scope here. From the standard equations of the parabolas, Fig11. , we have the following observations: .

Parabola is symmetric with respect to the axis of the parabola.If the equation has a y term, then the axis of symmetry is along the x -axis and if the equation has an x term, then the axis of symmetry is along the y -axis. . When the axis of symmetry is along the x -axis the parabola opens to the (a) right if the coefficient of x is positive, (b) left if the coefficient of x is negative. .

When the axis of symmetry is along the y -axis the parabola opens (c) upwards if the coefficient of y is positive. (d) downwards if the coefficient of y is negative. CONIC SECTIONS . .

Latus rectum Definition Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola (Fig11. ). To find the Length of the latus rectum of the parabola y = ax (Fig . ).

By the definition of the parabola, AF = AC. But AC = FM = a Hence AF = a . And since the parabola is symmetric with respect to x -axis AF = FB and so AB = Length of the latus rectum = a . Fig .

Fig . Example Find the coordinates of the focus, axis, the equation of the directrix and latus rectum of the parabola y = x . Solution The given equation involves y , so the axis of symmetry is along the x -axis. The coefficient of x is positive so the parabola opens to the right.

Comparing with the given equation y = ax , we find that a = . Thus, the focus of the parabola is ( , ) and the equation of the directrix of the parabola is x = – (Fig . ). Length of the latus rectum is a = × = .

Fig . MATHEMATICS Example Find the equation of the parabola with focus ( , ) and directrix x = – . Solution Since the focus ( , ) lies on the x -axis, the x -axis itself is the axis of the parabola. Hence the equation of the parabola is of the form either y = ax or y = – ax .

Since the directrix is x = – and the focus is ( , ), the parabola is to be of the form y = ax with a = . Hence the required equation is y = ( ) x = x Example Find the equation of the parabola with vertex at ( , ) and focus at ( , ). Solution Since the vertex is at ( , ) and the focus is at ( , ) which lies on y -axis, the y -axis is the axis of the parabola . Therefore, equation of the parabola is of the form x = ay .

thus, we have x = ( ) y , i.e., x = y. Example Find the equation of the parabola which is symmetric about the y -axis, and passes through the point ( ,– ). Solution Since the parabola is symmetric about y -axis and has its vertex at the origin, the equation is of the form x = ay or x = – ay , where the sign depends on whether the parabola opens upwards or downwards. But the parabola passes through ( ,– ) which lies in the fourth quadrant, it must open downwards.

Thus the equation is of the form x = – ay . Since the parabola passes through ( ,– ), we have = – a (– ), i.e., a = Therefore, the equation of the parabola is x = − y , i.e., x = – y . EXERCISE . In each of the following Exercises to , find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.

x = – y . y = x . x = – y In each of the Exercises to , find the equation of the parabola that satisfies the given conditions: CONIC SECTIONS Fig . Fig .

Fig . We denote the length of the major axis by a , the length of the minor axis by b and the distance between the foci by c . Thus, the length of the semi major axis is a and semi-minor axis is b (Fig11. ).

. Focus ( , ); directrix x = – . Focus ( ,– ); directrix y = . Vertex ( , ); focus ( , ) .

Vertex ( , ); focus (– , ) . Vertex ( , ) passing through ( , ) and axis is along x -axis. . Vertex ( , ), passing through ( , ) and symmetric with respect to y -axis.

. Ellipse Definition An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. The two fixed points are called the foci (plural of ‘ focus ’) of the ellipse (Fig11. ).