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the nearest point on the circle x^2+y^2-6x+4y-12=0 from( -5,4) is..? answer is (-1,1) can anyone answer this plzz...
Most Upvoted Answer
the nearest point on the circle x^2+y^2-6x+4y-12=0 from( -5,4) is..? a...
Given Data:
The equation of the circle is x^2 + y^2 - 6x + 4y - 12 = 0
The point is (-5,4)
Approach:
To find the nearest point on the circle from the given point, we need to first find the center of the circle and then find the point on the circle that is closest to the given point.
Finding the Center of the Circle:
To find the center of the circle, we need to rewrite the equation of the circle in the standard form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
x^2 + y^2 - 6x + 4y - 12 = 0
(x^2 - 6x) + (y^2 + 4y) = 12
Complete the square for x and y:
(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4
(x - 3)^2 + (y + 2)^2 = 25
The center of the circle is (3, -2) and the radius is 5.
Finding the Nearest Point:
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Calculate the distance from (-5, 4) to the center (3, -2):
Distance = sqrt((-5 - 3)^2 + (4 - (-2))^2) = sqrt((-8)^2 + (6)^2) = sqrt(64 + 36) = sqrt(100) = 10
As the radius of the circle is 5, the nearest point on the circle from (-5, 4) will be at a distance of 5 units from the center.
The point at a distance of 5 units from the center (3, -2) is (-1, 1).
Therefore, the nearest point on the circle from (-5, 4) is (-1, 1).
The equation of the circle is x^2 + y^2 - 6x + 4y - 12 = 0
The point is (-5,4)
Approach:
To find the nearest point on the circle from the given point, we need to first find the center of the circle and then find the point on the circle that is closest to the given point.
Finding the Center of the Circle:
To find the center of the circle, we need to rewrite the equation of the circle in the standard form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
x^2 + y^2 - 6x + 4y - 12 = 0
(x^2 - 6x) + (y^2 + 4y) = 12
Complete the square for x and y:
(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4
(x - 3)^2 + (y + 2)^2 = 25
The center of the circle is (3, -2) and the radius is 5.
Finding the Nearest Point:
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Calculate the distance from (-5, 4) to the center (3, -2):
Distance = sqrt((-5 - 3)^2 + (4 - (-2))^2) = sqrt((-8)^2 + (6)^2) = sqrt(64 + 36) = sqrt(100) = 10
As the radius of the circle is 5, the nearest point on the circle from (-5, 4) will be at a distance of 5 units from the center.
The point at a distance of 5 units from the center (3, -2) is (-1, 1).
Therefore, the nearest point on the circle from (-5, 4) is (-1, 1).
Community Answer
the nearest point on the circle x^2+y^2-6x+4y-12=0 from( -5,4) is..? a...
Answer come from only (-1,1) not from (-5,4)
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