Ola, I'm not understanding this question, any assistance? Thank you very much.
Find the distance of the line y = 2x + 6 from the circle x^2 + y^2 = 0.5 Find
the point of the line that is closest to the circle.
[Hint: a point on the line is closest to the circle if it is closest to the centre of the circle.]
The distance is ______________
the line has slope 2.
So, a perpendicular to the line will have slope -1/2, which will be the slope of the radius.
A line through (0,0) with slope -1/2 is y = -1/2 x.
This line intersects the circle at (-2/√10 , 1/√10). Now just find the distance between that point and the line.
You can also follow the hint and minimize the distance formula directly.
Thank you very much, Steve.
Hello! I can definitely help you with that. To find the distance between the line and the circle, we need to find the closest point on the line to the center of the circle.
Let's start by analyzing the given equation. The equation of the line is y = 2x + 6, and the equation of the circle is x^2 + y^2 = 0.5.
To find the point on the line that is closest to the circle, we can use the concept of perpendicular distance. We know that a line perpendicular to the given line will pass through the center of the circle.
The given line has a slope of 2, so a perpendicular line will have a slope of -1/2 (the negative reciprocal). Thus, the equation of the perpendicular line passing through the center of the circle can be written as:
y = (-1/2)x + c
Now, we need to find the value of c. Since the line is passing through the center of the circle, we can substitute the center coordinates (0,0) into the equation:
0 = (-1/2)(0) + c
0 = c
Therefore, the equation of the perpendicular line is y = (-1/2)x.
To find the point of intersection between the given line and the perpendicular line, we can equate the two equations:
2x + 6 = (-1/2)x
Simplifying the equation, we can solve for x:
2x + (1/2)x = -6
(5/2)x = -6
x = -12/5
Substitute this value of x back into either equation to find the corresponding y-value:
y = 2(-12/5) + 6
y = -24/5 + 30/5
y = 6/5
Therefore, the point on the line that is closest to the center of the circle is (-12/5, 6/5).
To find the distance between this point and the center of the circle, we can use the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's substitute the coordinates of the center of the circle (0,0) and the coordinates of the point on the line (-12/5, 6/5) into the formula:
d = sqrt((0 - (-12/5))^2 + (0 - 6/5)^2)
d = sqrt((12/5)^2 + (-6/5)^2)
d = sqrt(144/25 + 36/25)
d = sqrt(180/25)
d = sqrt(36/5)
d = 6/√5
Therefore, the distance of the line y = 2x + 6 from the circle x^2 + y^2 = 0.5 is 6/√5.