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Find out the equation of the circle which passes through a (2, - 1) and is tangent to the line x + y = 1, and the center of the circle is on the line y = - 2x. (I) find out the standard equation of the circle; (II) find out the chord length AB of the intersection of the circle in (I) and the line 3x + 4Y = 0
(1) Since the center of the circle C is on the straight line y = - 2x, let the center of the circle be C (a, - 2A). Then the distance D ′ = | a − 2A − 1 | 2 = | a + 1 | 2 from point C to the straight line x + y = 1. According to the title, D ′ = | AC |, then | a + 1 | 2 = (A-2) 2 + (- 2A + 1) 2, | a2-2a + 1 = 0 | a = 1
RELATED INFORMATIONS
- 1. When a circle passes through a (2,1) point and is tangent to a straight line X-Y = 0, and the center of the circle is on 2x-y = 0, the standard equation of the circle is obtained
- 2. As shown in the figure, in the rectangular coordinate system, point a is a point on the image of inverse scale function Y1 = KX, the positive half axis of ab ⊥ X axis is at point B, and C is the midpoint of OB; the image of primary function y2 = ax + B passes through two points a and C, and the Y axis is at point d (0, - 2), if s △ AOD = 4 (1) (2) observe the image, please point out the value range of X on the right side of y-axis when Y1 > Y2
- 3. As shown in the figure, in O, the chord AB = AC = 10, the chord ad intersects BC with E, AE = 4, and the length of ad is calculated
- 4. As shown in the figure, AB and AC are the chords of ⊙ o, ad ⊥ BC at D, intersection of ⊙ o at F, AE and diameter of ⊙ O. what is the relationship between the two chords be and the size of CF? Explain the reason
- 5. It is known that: as shown in the figure, AB is the diameter of ⊙ o, AC is the chord, CD ⊥ AB at D. if AE = AC, be intersects ⊙ o at point F, CF and de are connected
- 6. As shown in the figure, AB and AC are the chords of ⊙ o, ad ⊥ BC at D, intersection of ⊙ o at F, AE and diameter of ⊙ O. what is the relationship between the two chords be and the size of CF? Explain the reason
- 7. If ad ^ 2 = AE * AC, CD = CB
- 8. As shown in the figure, AB is the diameter of the circle O, ad = De, AE and BD intersect with point C, then there are several angles equal to angle BCE in the figure
- 9. As shown in the figure, AB is a semicircle, O is the midpoint of AB, C and D are on AB, and ad ‖ OC connects BC and BD. if CD = 62 °, what is the degree of ad? （ ） A. 56B. 58C. 60D. 62
- 10. In graph o, the strings AB and CD intersect at E. if arc ad = arc AC, we prove that AC squared = AE multiplied by ab
- 11. Given that the circle passes through the point P (2, - 1) and is tangent to the straight line X-Y = 1, and its center is on the straight line y = - 2x, the equation of the circle is obtained
- 12. Given a circle passing through point P (2, - 1), the center of the circle is on L1: y + 2x = 0 and tangent to the line L2: x-y-1 = 0, the equation of the circle is obtained
- 13. A circle passes through point P (2, - 1) and is tangent to the straight line X-Y = 1. The center of the circle is on the straight line y = - 2x
- 14. If the circle C passes through (2, - 1) and is tangent to the line x-y-1 = 0, and the center of the circle is on the line y = - 2x, then the standard equation of the circle C is
- 15. It is known that the center coordinate of the circle is (- 1,2) and tangent to the line 2x + Y-5 = 0 at point M. the standard equation of the circle is obtained
- 16. Given that the circle C passes through point a (2, - 1), the center of the circle is on the line 2x + y = 0 and tangent to the line x + y = 0, then the standard equation of circle C is
- 17. Given the circle m, the center of the circle is on the line 2x + y = 0, and tangent to the line x + Y-1 = 0 at the point a (2, - 1), find the standard equation of the circle m
- 18. Find the standard equation of the circle whose center is on the straight line 2x-y-3 = 0 and passes through points (5,2) and (3, - 2)
- 19. The standard equation of a circle passing through points a (5,2) and B (3, - 2) with the center of the circle on the line 2x-y-3 = 0 I will find out the straight line L through AB, then find out the equation of AB vertical bisector, and then combine 2x-y-3 = 0 to find the center of the circle. But the answer is wrong. Is this method wrong?
- 20. It is known that the center of circle C is on the straight line 2x-y-3 = 0, and the standard equation of circle C can be obtained through points a (5,2), B (3,2)