Final answer:
The minimum length of the chord of the given circle that passes through the point (1,0) is calculated by finding the perpendicular distance from the point to the line connecting the center of the circle to the point. After completing the square to find the center and radius of the circle, and using the Pythagorean theorem, the minimum chord length is determined to be 4 units.
Step-by-step explanation:
The student is asking to find the minimum length of the chord of the circle x²+y²+2x+2y−7=0 which passes through the point (1,0). To solve this, we first need to complete the square for both x and y to find the center and the radius of the circle.
The given equation can be rewritten as:
(x + 1)² - 1 + (y + 1)² - 1 - 7 = 0
By rearranging, we get:
(x + 1)² + (y + 1)² = 9
Thus, the circle has a center at (-1, -1) and a radius of 3.
Now, the minimum length chord through (1,0) will be perpendicular to the radius of the circle at that point. Using the formula for the distance from a point to a line (the perpendicular distance from the point (1,0) to the line connecting the center and (1,0)), we will find the length of this perpendicular segment. This length will be multiplied by 2 to get the minimum chord length, as it will be a diameter section.
The distance (d) from the point (1,0) to the center (-1,-1) is:
d = √((1 + 1)² + (0 + 1)²)
After calculating, we have:
d = √((2)² + (1)²) = √5)
The radius is r = 3, so the length of the perpendicular segment from (1,0) to the diameter is √(r² - d²) = √(9 - 5) = √4 = 2. Therefore, the minimum chord length is twice this value: 4 units.