Respuesta :
Final answer:
The locus of the point P, if the triangle PAB is equilateral, is x²+y²=8. To find this, we can start by finding the coordinates of point A and point B using the circle equation, and then use the fact that the triangle is equilateral to find the coordinates of point P.
option b is the correct answer
Explanation:
The locus of the point P, if the triangle PAB is equilateral, is given by the equation x²+y²=8 (option B).
To solve this problem, we can start by finding the coordinates of point A and point B using the circle equation. Substituting x=2 and y=0 into the equation x²+y²=4, we find that point A is (2,0).
Similarly, substituting x=-2 and y=0, point B is (-2,0).
Now, we can use the fact that the triangle PAB is equilateral to find the coordinates of point P.
Since PA and PB are tangents to the circle, they are perpendicular to the radius of the circle at points A and B.
Therefore, the distance from the center of the circle to point P is also 4.
Since the radius of the circle is 2, the coordinates of point P are (0,4).
Plugging these coordinates into the equation of a circle, we find that 0²+4²=8.
Therefore, the locus of point P is x²+y²=8.
Final answer:
The locus of the point P if the triangle PAB is equilateral is x²+y²=16.
Explanation:
The locus of the point P if the triangle PAB is equilateral can be found by analyzing the tangents drawn to the circle x²+y²=4. Let's start by finding the equation of the tangent. The equation to the trajectory can be written as: (x − A)² = -4a(y – B), where A and B are constants related to the coordinates of the tangent points.
Next, we can use the information about the equilateral triangle to find the relationship between A and B. From the expressions given, we can determine that A = 4x². Using this relationship, we can rewrite the equation of the trajectory as (x − 4x²)² = -4a(y – B).
Finally, we can simplify the equation and find the locus of the point P. By expanding the equation and combining like terms, we get x² - 8x² + 16x² = -16y + 4B. Simplifying further, we get -x² + 16x² = -16y + 4B, which simplifies to x²+y²=16. Therefore, the locus of the point P is x²+y²=16.
