Some Selected Problems in Analytic Geometry I Russelle Guadalupe July 26, 2012
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The Straight Line ax + by + c = 0 1. Find the equation of the line l passing through M (0, 1) such that the lines l, l1 : x − 7y + 17 = 0, l2 : x + y − 5 = 0 form an isosceles triangle with base l. 2. Find the equation of the line l passing through P (2, −1) such that the lines l, l1 : 2x−y +5 = 0, l2 : 3x+6y −7 = 0 form a isosceles right triangle. 3. Let l1 : 3x + y + 5 = 0 and l2 : 3x + y + 1 = 0. Suppose that a line √ l passing through I(1, −2) meets the lines l1 , l2 at points A, B, respectively. Find the equation of l if AB = 4 2. 4. A line passing through M (3, 1) meets the x-axis at A and y-axis at B. Find the equation of the line for which OA + 3OB is minimum. 5. A line passing through M (1, 2) meets the x-axis at A and y-axis at B. Find the equation of the line for which 4 9 + is minimum. 2 OA OB 2 6. A line passing through M (3, 1) meets the x-axis at B and y-axis at C. Find the equation of the line so that ABC is an isosceles right triangle with a right angle at A(2, −2). 7. Find the equation of the line l passing through M (2, 1) such that the area of the triangle formed by l and the coordinate axes is 4. 8. Find the equation of the line l passing through A(−2, 1) such that the angle α formed by l and the line 2x−y+3 = 1 0 has cos α = √ . 10 9. Find the equation of the line l passing through A(2, 1) such that the angle formed by l and the line 2x+3y +4 = 0 is 45◦ .
0 10. Consider the line l : 2x − y − 2 = 0 and the point √ I(1, 1). Find the equation of the line l if the angle formed by 0 ◦ 0 l and l is 45 and the distance from I to l is 10.
11. Let the lines l1 : 3x + y + 2 = 0 and l2 : x − 3y + 4 = 0 meet at A. Suppose that the line l through M (0, 2) meets 1 1 l1 at B and l2 at C. Find the equation of l if + is minimum. AB 2 AC 2 12. Find a point M on the line x − 3y − 4 = 0 and a point N on the circle x2 + y 2 − 4y = 0 such that the midpoint of M N is (3, 1). 13. Consider the point A(1, 1). Find a point B on the line l : 2x + 3y + 4 = 0 such that the angle formed by AB and l is 45◦ . 14. Given the point N (3, 4), the origin O and the line l : x − 3y − 6 = 0, find a point M on l such that the area of triangle OM N is 15. 15. Given the point A(0, 2) and the line l : x − 2y + 2 = 0, find points B and C on l such that ∠ABC = 90◦ and AB = 2BC. 16. Given the parallel lines l1 : x + y − 3 = 0, l2 : x + y − 9 = 0 and the point A(1, 4), find a point B on l1 and C on l2 such that ∠BAC = 90◦ and AB = AC. 17. Let A(0, 1) and B(2, −1) be points on the coordinate plane, and let P be the intersection point of the lines l1 : (m − 1)x + (m − 2)y + 2 − m = 0 l2 : (2 − m)x + (m − 1)y + 3m − 5 = 0 Find the value of m if P A + P B is maximum. 1
18. Given the points A(−1, 2) and B(3, 4), find a point M on the line x − 2y − 2 = 0 such that 2M A2 + M B 2 is minimum. 19. Given the points A(1, 0) and B(2, 1), find a point M on the line 2x − y + 3 = 0 such that M A + M B is minimum. √ 20. Find the equation of the line passing through (1, 5) and has a distance 2 10 from (6, 0).
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The Circle (x − h)2 + (y − k)2 = r2 1. Suppose that the line 2x − y − 5 = 0 intersects the circle x2 + y 2 − 20x + 50 = 0 at two points A and B. Find the equation of the circle passing through A, B and C(1, 1). 2. Suppose that the centroid G1 of triangle ABC with points A(2, −3), B(3, −2) and area 3x − y − 8 = 0. Find the equation of the circumcircle2 of triangle ABC.
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3. Determine the equation of the circle with its center lying on the line 2x + y − 3 = 0 and tangent to the lines 3x + 4y + 5 = 0 and 4x + 3y + 2 = 0. 4. Determine the equation of the circle with its center lying on the line x + 3y + 8 = 0 and tangent to the line 3x − 4y + 10 = 0 at the point (−2, 1). 5. Determine the equation of the circle which is tangent to the line 4x − 3y + 3 = at a point with y-coordinate 9 and tangent to the line 3x − 4y − 31 = 0. 6. Determine the equation of the circle passing through (2, −1) and tangent to the coordinate axes. 7. Determine the equation of the circle with its center lying on the line 2x − y − 4 = 0 and tangent to the coordinate axes. 8. Determine the equation of the circle passing through the points (−1, 1), (3, 3) and tangent to the line 3x−4y+8 = 0. 9. Determine the equation of the circle passing through the point (1, 1), with its center lying on the line x−y +6 = 0 and tangent to 3x − 4y + 2 = 0. √ 2 10 10. Determine the equation of the circle with radius , with its center lying on the line x + 2y − 3 = 0 and 5 tangent to the line x + 3y − 5 = 0. √ 11. The circle C : x2 + y 2 + 4 3x − 4 = 0 meets the y-axis at the point A. Find the equation of the circle C 0 with radius 2 and externally tangent to C at A. 12. A circle C 0 with center √ (5, 1) meets a circle C : x2 + y 2 − 2x + 4y + 2 = 0 at two points A and B. Find the equation of C 0 if AB = 3. 13. Given the circle C : (x − 1)2 + (y − 2)2 = 4 with center I and a point K(3, 4), determine the equation of the circle with center K that cuts C at two points A and B such that the area of triangle IAB is maximum. 1 3 14. Find the equation of the incircle of a triangle with vertices A(−2, 3), B , 0 , C(2, 0). 4 15. Determine the equation of a circle with its center on the line x − y − 1 = 0 and externally tangent to the circles (x − 3)2 + (y + 4)2 = 8 and (x + 5)2 + (y − 4)2 = 32. 16. Given the points A(3, −7), B(9, −5), C(−5, 9) and M (−2, −7), find the equation of the line through M and tangent to the circumcircle of 4ABC. 17. Given the circle C : x2 + y 2 + 2x = 0, find the equation of the line tangent to C and has the angle 30◦ with the y-axis. 18. Given the circle C : x2 + y 2 − 6x − 2y + 5 = 0 and the line l : 3x + y − 3 = 0, find the equation of the line tangent to C and has the angle 45◦ with l. 19. Repeat Problem 18 for the circle C : (x − 1)2 + (y − 1)2 = 10 and the line l : 2x − y − 2 = 0. 1 The centroid G of a triangle is the intesection point of the medians, while a median is a line segment joining the vertex and the midpoint of its opposite side of a triangle. 2 The circumcircle of a triangle is a circle passing through the vertices of that triangle. 3 The incircle of a triangle is a circle whose tangents are the sides of a triangle.
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20. Find the equations of the common external and internal tangents to the circles x2 + y 2 − 2x − 2y − 2 = 0 and x2 + y 2 − 8x − 2y + 16 = 0. 21. Repeat Problem 20 for the circles (x − 2)2 + (y − 3)2 = 2 and (x − 1)2 + (y − 2)2 = 8. 22. Repeat Problem 20 for the circles x2 + y 2 − 2y − 3 = 0 and x2 + y 2 − 8x − 8y + 28 = 0. 23. Repeat Problem 20 for the circles x2 + y 2 − 4y − 5 = 0 and x2 + y 2 − 6x + 8y + 16 = 0. 24. Determine the value of the parameter m so that the circle x2 + y 2 = 1 is internally tangent to the circle x2 + y 2 − 2(m + 1)x + 4my − 5 = 0. 1 25. Suppose that a line tangent to the circle (x − 1)2 + y 2 = meets another circle (x − 2)2 + (y − 2)2 = 4 at two 2 √ points M and N such that M N = 2 2. Find the equation of the line. 26. Find a point M on the y-axis such that the angle formed by the tangents from M to the circle x2 +y 2 −6x+5 = 0 is 60◦ . 27. Find a point M on the line x + 2y − 12 = 0 such that the angle formed by the tangents from M to the circle x2 + y 2 − 4x − 2y = 0 is 60◦ . 28. Suppose that the tangents from the point M on the circle x2 + y 2 − 18x − 6y + 65 = 0 to the second circle 24 . C : x2 + y 2 = 9 touch C at points A and B. Find the point M if AB = 5 29. The tangents from a point M on the line x − y + 1 = 0 to the circle C : (x − 1)2 + (y + 2)2 = 4 touch C at points A and B. Find M if the midpoint of AB is (1, −1). 30. A line passing through M (7, 3) intersects the circle (x − 1)2 + (y + 1)2 = 25 at two distinct points A and B. Find the equation of the line if M A = 3M B. 2 2 31. A line passing through M (5, 2) intersects √ the circle x + y − 4x − 8y − 5 = 0 at two distinct points A and B. Find the equation of the line if AB = 5 2.
32. A line perpendicular to 3x − 4y + 10 = 0 meets the circle (x + 4)2 + (y − 3)2 = 25 at two points A and B. Find the equation of the line if AB = 6. 33. Determine the equation of the line through (0, 2) which intersects the circle x2 + y 2 − 2x − 2y − 3 = 0 at two distinct points A and B such that AB has minimum length. 34. Suppose that the line through M (1, −8) meets the circle x2 + y 2 + 4x − 6y + 9 = 0 with center C at two points A and B. Find the equation of the line if 4ABC has maximum area. 35. Let A be the intersection point with positive y-coordinate of the circles ω1 : x2 +y 2 = 13 and ω2 : (x−6)2 +y 2 = 25. A line through A meets ω1 at B and ω2 at C. Find the equation of the line if A is the midpoint of BC. 36. A line mx + 4y = 0 meets the circle x2 + y 2 − 2x − 2my + m2 − 24 = 0 with center C at two points A and B. Find the value of m if the area of 4ABC is 12. 37. Find the value of m for which the line x + y + m = 0 meet the circle x2 + y 2 = 1 at points A and B such that 4OAB has maximum area, where O is the origin. 38. A line x + my − 2m + 3 = 0 meets the circle x2 + y 2 + 4x + 4y + 6 = 0 with center C at two points A and B. Find the value of m if the area of 4ABC is maximum. 39. A line through M (1, −8) meets the circle x2 + y 2 + 4x − 6y + 9 = 0 with center C at two points A and B. Find the equation of the line if the area of 4ABC is maximum. 40. The line x − 5y − 2 = 0 intersects the circle x2 + y 2 + 2x − 4y − 8 = 0 at two points A (with positive x-coordinate) and B. Find a point C on the circle such that ∠ABC = 90◦ . 41. The line 2x − 3y − 1 = 0 intersects the circle x2 + y 2 + 2x − 4y − 8 = 0 at two points A and B. Find a point C on the circle such that the area of 4ABC is maximum. 42. Given the circle x2 + y 2 − 2x − 4y − 5 = 0 and a point A(0, −1), find two other points B, C on the circle such that ABC is an equilateral triangle. 43. Given the circle (x − 3)2 + (y − 4)2 = 35 and a point A(5, 5), find two other points B, C on the circle such that AB = AC and ∠BAC = 90◦ .
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8 44. Let A 1, − , B(3, 0) be two points on the coordinate plane. Find a point M on the circle x2 + y 2 = 4 such 3 20 that [M AB] = . ([M AB] denotes the area of triangle M AB.) 3 45. Find a point M on the circle x2 + y 2 + 2x − 6y + 9 = 0 and a point N on the line 3x − 4y + 5 = 0 such that M N has minimum length.
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The Conics 1. Given the ellipse
y2 x2 + = 1 and its two foci F1 , F2 , find a point M on the ellipse such that ∠F1 M F2 = 120◦ . 100 25
2. Given the ellipse 5x2 + 16y 2 = 80 and two points A(−5, −1), B(−1, 1), find a point M on the ellipse such that the area of 4M AB is maximum, and find that maximum area. y2 x2 + = 1 and two points A(3, −2), B(−3, 2), find a point C on the ellipse such that the 3. Given the ellipse 9 4 area of 4ABC is maximum. x2 y2 4. A line through M (1, 1) cuts the ellipse + = 1 at two points A and B. Find the equation of the line if M 25 9 is the midpoint of AB. 5. Given the parabola y 2 = x and a point I(0, 2), find two points M and N on the parabola such that IM = 4IN .
Russelle Guadalupe 19F G. Marcelo Street, Maysan Valenzuela City 1440 Philippines
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