Askiitians Tutor Team
Last Activity: 9 Months ago
To find the slope of the line passing through point P(2,3) and intersecting the line x + y = 7 at a distance of 4 units from P, we can follow these steps:
First, find the equation of the line x + y = 7 in slope-intercept form (y = mx + b), where m is the slope of the line.
Then, find the point(s) where the line from P intersects the line x + y = 7 and is 4 units away from P.
Finally, calculate the slope of the line passing through P and the intersection point(s).
Let's start with step 1:
Find the equation of the line x + y = 7 in slope-intercept form:
x + y = 7
y = -x + 7
So, the slope of the line x + y = 7 is -1.
Now, let's move to step 2:
Find the point(s) where the line from P intersects the line x + y = 7 and is 4 units away from P.
Let the intersection point be Q(x, y).
We know that the distance between P(2,3) and Q(x, y) is 4 units. We can use the distance formula:
Distance formula: √((x2 - x1)^2 + (y2 - y1)^2) = 4
Substitute the coordinates of P(2,3):
√((x - 2)^2 + (y - 3)^2) = 4
Now, let's move to step 3:
Calculate the slope of the line passing through P and the intersection point(s).
We already found that the slope of the line x + y = 7 is -1. To find the slope of the line passing through P and Q, we need to find the coordinates of Q.
From the distance formula, we have:
√((x - 2)^2 + (y - 3)^2) = 4
Simplify:
(x - 2)^2 + (y - 3)^2 = 16
Now, we need to find the coordinates of Q that satisfy this equation and are also on the line x + y = 7 (y = -x + 7).
Substitute y = -x + 7 into the equation:
(x - 2)^2 + (-x + 7 - 3)^2 = 16
Simplify and solve for x:
(x - 2)^2 + (-x + 4)^2 = 16
Expand and simplify further:
x^2 - 4x + 4 + x^2 - 8x + 16 = 16
Combine like terms:
2x^2 - 12x + 4 = 16
2x^2 - 12x - 12 = 0
Divide both sides by 2:
x^2 - 6x - 6 = 0
Now, we can solve this quadratic equation for x using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -6, and c = -6:
x = (6 ± √((-6)^2 - 4(1)(-6))) / (2(1))
x = (6 ± √(36 + 24)) / 2
x = (6 ± √60) / 2
x = 3 ± √15
So, there are two possible x-values for Q: x = 3 + √15 and x = 3 - √15.
Now, we can find the corresponding y-values by using y = -x + 7:
For x = 3 + √15:
y = -(3 + √15) + 7
y = -3 - √15 + 7
y = 4 - √15
For x = 3 - √15:
y = -(3 - √15) + 7
y = -3 + √15 + 7
y = 4 + √15
Now that we have the coordinates of Q, we can calculate the slope of the line passing through P(2,3) and Q:
For the point (3 + √15, 4 - √15):
Slope = (4 - √15 - 3) / (3 + √15 - 2)
Slope = (1 - √15) / (1 + √15)
For the point (3 - √15, 4 + √15):
Slope = (4 + √15 - 3) / (3 - √15 - 2)
Slope = (1 + √15) / (1 - √15)
Now, let's simplify both possibilities:
Slope 1: (1 - √15) / (1 + √15)
Slope 2: (1 + √15) / (1 - √15)
Notice that Slope 1 is the negative reciprocal of Slope 2.
So, the two possible slopes for the line passing through P and intersecting the line x + y = 7 at a distance of 4 units from P are:
Slope 1: (1 - √15) / (1 + √15)
Slope 2: (1 + √15) / (1 - √15)
Now, let's simplify these slopes further:
Slope 1:
Multiply both the numerator and denominator by (1 - √15):
Slope 1 = [(1 - √15)(1 - √15)] / [(1 + √15)(1 - √15)]
Slope 1 = (1 - 2√15 + 15) / (1 - 15)
Slope 1 = (16 - 2√15) / (-14)
Simplify the negative sign:
Slope 1 = -(8 - √15) / 7
Slope 2:
Multiply both the numerator and denominator by (1 + √15):
Slope 2 = [(1 + √15)(1 + √15)] / [(1 - √15)(1 + √15)]
Slope 2 = (1 + 2√15 + 15) / (1 - 15)
Slope 2 = (16 + 2√15) / (-14)
Simplify the negative sign:
Slope 2 = -(8 + √15) / 7
So, the two possible slopes for the line passing through P and intersecting the line x + y = 7 at a distance of 4 units from P are:
Slope 1: -(8 - √15) / 7
Slope 2: -(8 + √15) / 7
None of these options exactly matches the choices (a), (b), (c), or (d). It's possible that there may be a mistake in the answer choices or the question itself. Please double-check the question and answer choices for accuracy.
