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Formulas and definitions of circle
formula and definitions of ellipse
formula and definition of hyperola
In each of the following Exercises 1 to 4, find the equation of the circle with 1. centre (0,2) and radius 2 ...
find the equation of the circle with . centre (–a, –b) and radius root (a2 - b2)
9.find the centre and radius of the circles 2x 2 + 2y – x = 0 2
10. Find the equation of the circle passing through the points (4,1) and (6,5) and whose centre is on the line 4x + y = 16.
12. Find the equation of the circle with radius 5 whose centre lies on x-axis and passes through the point (2,3).
13. Find the equation of the circle passing through (0,0) and making intercepts a and b on the coordinate axes.
14. Find the equation of a circle with centre (2,2) and passes through the point (4,5).
15. Does the point (–2.5, 3.5) lie inside, outside or on the circle x2 + y2 = 25?
In each of the following Exercises 1 to 6, find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.
In each of the Exercises 7 to 12, find the equation of the parabola that satisfies the given conditions: 7. Focus (6,0); directrix x = – 6 8. Focus (0,–3); directrix y = 3 9. Vertex (0,0); focus (3,0) 10. Vertex (0,0); focus (–2,0)
12. Vertex (0,0), passing through (5,2) and symmetric with respect to y-axis.
In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.find the minor axis the eccentricity and the length of the latus rectum of the ellipse
10. Find the equation for the ellipse whose vertex and foci are given Vertices (± 5, 0), foci (± 4, 0)
13. Ends of major axis (± 3, 0), ends of minor axis (0, ± 2) find the equation for the ellipse that satisfies the given conditions
14. Ends of major axis (0, ±), ends of minor axis (± 1, 0) find the equation for the ellipse that satisfies the given conditions
15. Length of major axis 26, foci (± 5, 0) find the equation for the ellipse that satisfies the given conditions:
17. Foci (± 3, 0), a = 4find the equation for the ellipse that satisfies the given conditions:
18. b = 3, c = 4, centre at the origin; foci on a x axis.find equation for ellipse whose centre at major axis on y-axis and passes through points and
20 . Major axis on the x-axis and passes through the points (4,3) and (6,2).
Q 1- 6 In each of the Exercises 1 to 6, find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.
Q 7 - 11. In each of the Exercises 7 to 15, find the equations of the hyperbola satisfying the given conditions. Vertices (± 2, 0), foci (± 3, 0)
Q 12 - 13 Foci (± 3, 0), the latus rectum is of length 8. 13. Foci (± 4, 0), the latus rectum is of length 12 5
14. vertices (± 7,0), e = . 15. Foci (0, ± 10), passing through (2,3)
Example 17 The focus of a parabolic mirror as shown in Fig 11.33 is at a distance of 5 cm from its vertex. If the mirror is 45 cm deep, find
Ex - 18 A beam is supported at its ends by2 Fig 11.33 supports which are 12 metres apart. Since the load is concentrated at its centre, there is a deflection of 3 cm at the centre and the deflected beam is in the shape of a parabola. How far from the centre is the deflection 1 cm?
Example 19 A rod AB of length 15 cm rests in between two coordinate axes in such a way that the end point A lies on x-axis and end point B lies on y-axis. A point P(x, y) is taken on the rod in such a way that AP = 6 cm. Show that the locus of P is an ellipse.
1. If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?
3. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.
4. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.
5. A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.
6. Find the area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends of its latus rectum.
7. A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.
8. An equilateral triangle is inscribed in the parabola y = 4 ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
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