what are the equation of a line through (8,-3) and passing distance square root of 10 from (3,2)
To be a "distance of √10 units away from (3,2) means it has to be tangent to the circle
(x-3)^2 + (y-2)^2 = 10
or
x^2 + y^2 -6x - 4y = -3
let the point of contact be (a,b)
then
a^2 + b^2 - 6a - 4b = -3 (#1)
by slopes...
slope of radius to (a,b) = (b-2)/(a-3)
slope from (a,b) to (8,-3) = (b+3)/(a-8)
but these two lines are perpendicular, so ...
(b+3)/(a-8) = - (a-3)/(b-2)
which simplifies to
a^2 + b^2 - 11a + b = -18 ,(#2)
subtract #1 - #2
5a - 5b = 15
a - b = 3
a = b+3 (#3)
sub #3 into #2 to get it simplified to
b^2 - 2b - 3 = 0
(b-3)(b+1) = 0
b = 3 or b = -1
in #3
a= 6 or a = 2
so (a,b) is either (6,3) or (2,-1)
Now it is easy to find the equation of the two tangents.
Let me know what you got.
3x+y-21=0 and x+3y+1
To find the equation of a line passing through the point (8, -3) and a given distance from another point (3, 2), we can follow these steps:
Step 1: Find the slope of the line passing through the two points.
- The slope can be calculated using the formula: m = (y2 - y1) / (x2 - x1).
- Let's use the points (8, -3) and (3, 2) to find the slope.
- Plugging in the values, we get: m = (2 - (-3)) / (3 - 8) = (2 + 3) / (3 - 8) = 5 / (-5) = -1.
Step 2: Determine the distance between the point (8, -3) and the given point (3, 2).
- We need to find the distance in order to determine the equation of the line that passes through the given distance from the point (3, 2).
- The distance between two points can be calculated using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
- Plugging in the values, we get: d = sqrt((8 - 3)^2 + (-3 - 2)^2) = sqrt(5^2 + (-5)^2) = sqrt(25 + 25) = sqrt(50) = sqrt(2 * 5^2) = 5 * sqrt(2).
Step 3: Use the point-slope form of the equation to find the equation of the line.
- The point-slope form of the equation is: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
- Plugging in the values, we get: y - (-3) = -1(x - 8).
- Simplifying, we have: y + 3 = -x + 8.
- Rearranging the equation, we get: x + y = 5.
Therefore, the equation of the line passing through the point (8, -3) and a distance of sqrt(10) from (3, 2) is x + y = 5.