Analytic Geometry Problems

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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


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