Find the value of k such that the distance of 3x + 4y + 7 = 0 to (2, k) is 5.
To find the value of k that satisfies the condition, we need to consider the distance formula between a point and a line.
The equation of the line in standard form is 3x + 4y + 7 = 0. To determine the distance between this line and the point (2, k), we need to construct a perpendicular line that passes through the point (2, k) and intersects the original line.
First, let's find the slope of the given line. We can do this by rearranging the equation to solve for y:
3x + 4y + 7 = 0
4y = -3x - 7
y = (-3/4)x - 7/4
The given line has a slope of -3/4. For the perpendicular line to intersect the given line, it must have a slope that is the negative reciprocal of -3/4. Therefore, the slope of the perpendicular line is 4/3.
Now, we can use the point-slope form of a line to write the equation of the perpendicular line that passes through (2, k):
y - k = (4/3)(x - 2)
Next, we can rearrange this equation to slope-intercept form:
y - k = (4/3)x - (8/3)
y = (4/3)x - (8/3) + k
y = (4/3)x - (8/3) + (3k/3)
y = (4/3)x - (8/3) + (3k/3)
y = (4/3)x + (3k-8)/3
To find the point of intersection between the two lines, we can set the two equations equal to each other:
(-3/4)x - 7/4 = (4/3)x + (3k-8)/3
To simplify this equation, we can multiply each term by 12 to eliminate the fractions:
-9x - 21 = 16x + 4(3k-8)
-9x - 21 = 16x + 12k - 32
Now, we can combine like terms:
-9x - 16x = 12k - 32 + 21
-25x = 12k - 11
Finally, to find the value of k, we substitute the coordinates of the point (2, k) into the equation above, and the distance formula between the two points should equal 5:
√((2 - (-25x))^2 + (k - ((4/3)x + (3k-8)/3))^2) = 5
Simplifying and solving for k:
√((27x + 2)^2 + (k - ((4/3)x + (3k-8)/3))^2) = 5
Square both sides to eliminate the square root:
(27x + 2)^2 + (k - ((4/3)x + (3k-8)/3))^2 = 5^2
(27x + 2)^2 + (3k/3 - 4x/3 - 8/3)^2 = 25
This equation can be solved using algebraic manipulation to find the value of x and subsequently, the value of k.