16 find the coordinates of a point on the parabola y

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16. Find the coordinates of a point on the parabola y 2 = 8 x whose focal distance is 4. 17. Find the length of the line-segment joining the vertex of the parabola y 2 = 4 ax and a point on the parabola where the line-segment makes an angle θ to the x axis. 18. If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola. 19. If the line y = mx + 1 is tangent to the parabola y 2 = 4 x then find the value of [ Hint: Solving the equation of line and parabola, we obtain a quadratic equation and then apply the tangency condition giving the value of m ]. 20. If the distance between the foci of a hyperbola is 16 and its eccentricity is 2 then obtain the equation of the hyperbola. 21. Find the eccentricity of the hyperbola 9 y 2 – 4 x 2 = 36. 22. Find the equation of the hyperbola with eccentricity 3 2 and foci at (± 2, 0). - m . , Long Answer Type 23. If the lines 2 x – 3 y = 5 and 3 x – 4 y = 7 are the diameters of a circle of area 154 square units, then obtain the equation of the circle. 24. Find the equation of the circle which passes through the points (2, 3) and (4, 5) and the centre lies on the straight line y – 4 x + 3 = 0. 25. Find the equation of a circle whose centre is (3, –1) and which cuts off a chord of length 6 units on the line 2 x – 5 y + 18 = 0. [ Hint: To determine the radius of the circle, find the perpendicular distance from the centre to the given line.] 26. Find the equation of a circle of radius 5 which is touching another circle x 2 + y 2 – 2 x – 4 y – 20 = 0 at (5, 5). 27. Find the equation of a circle passing through the point (7, 3) having radius 3 units and whose centre lies on the line y = x – 1. 28. Find the equation of each of the following parabolas (a) Directrix x = 0, focus at (6, 0) (b) Vertex at (0, 4), focus at (0, 2) (c) Focus at (–1, –2), directrix x – 2 y + 3 = 0
204 EXEMPLAR PROBLEMS – MATHEMATICS 29. Find the equation of the set of all points the sum of whose distances from the points (3, 0) and (9, 0) is 12. 30. Find the equation of the set of all points whose distance from (0, 4) are 2 3 of their distance from the line y = 9. 31. Show that the set of all points such that the difference of their distances from (4, 0) and (– 4, 0) is always equal to 2 represent a hyperbola. 32. Find the equation of the hyperbola with (a) Vertices (± 5, 0), foci (± 7, 0) (b) Vertices (0, ± 7), e = 4 (c) Foci (0, ± 10 ), passing through (2, 3) 3 Objective Type Questions State Whether the statements in each of the Exercises from 33 to 40 are True or False. Justify 33. The line x + 3 y = 0 is a diameter of the circle x 2 + y 2 + 6 x + 2 y = 0. 34. The shortest distance from the point (2, –7) to the circle x 2 + y 2 – 14 x – 10 y – 151 = 0 is equal to 5. [ Hint: The shortest distance is equal to the difference of the radius and the distance between the centre and the given point.] 35. If the line lx + my = 1 is a tangent to the circle x 2 + y 2 = a 2 , then the point ( l lies on a circle. [ Hint: Use that distance from the centre of the circle to the given line is equal to radius of the circle.] 36. The point (1, 2) lies inside the circle x 2 + y 2 – 2 x + 6 y + 1 = 0. 37. The line lx + my + n = 0 will touch the parabola y 2 = 4 ax if ln = , m ) am 2 . 38. If P is a point on the ellipse 2 2 1 16 25 x y + = whose foci are S and S , then PS + PS = 8. 39. The line 2 x + 3 y = 12 touches the ellipse 2 2 2 9 4 x y + = at the point (3, 2). 40.
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