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The perpendicular bisector of the line segment joining P (1, 4) and Q(k, 3) has y-intercept –4. Then a possible value of k is[2008]
- a)1
- b)2
- c)–2
- d)– 4
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The perpendicular bisector of the line segment joining P (1, 4) and Q(...
Slope of 
∴ Slope of perpendicular bisector of PQ = ( k –1)
Also mid point of 
∴ Equation of perpendicular bisector is
⇒ 2y – 7 = 2(k –1) x –(k2 –1)
⇒ 2(k – 1)x – 2y + ( 8 – k2) = 0
∴ y-intercept 
⇒ 8 – k2 = –8 or k2 = 16 ⇒ k = ± 4
Most Upvoted Answer
The perpendicular bisector of the line segment joining P (1, 4) and Q(...
To find the perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3), we first need to find the midpoint of the line segment. The midpoint is the average of the x-coordinates and the average of the y-coordinates.
The x-coordinate of the midpoint is (1 + k) / 2 = (k + 1) / 2.
The y-coordinate of the midpoint is (4 + 3) / 2 = 7 / 2.
So, the midpoint is M((k + 1) / 2, 7 / 2).
The slope of the line segment joining P and Q is (3 - 4) / (k - 1) = -1 / (k - 1).
Since the perpendicular bisector is perpendicular to the line segment, its slope will be the negative reciprocal of the slope of the line segment. So, the slope of the perpendicular bisector is (k - 1) / 1.
Since the perpendicular bisector passes through the midpoint M, we can use the point-slope form of a line to find the equation of the perpendicular bisector.
y - (7 / 2) = (k - 1)(x - (k + 1) / 2).
To find the y-intercept, we can set x = 0 and solve for y:
y - (7 / 2) = (k - 1)(0 - (k + 1) / 2).
y - (7 / 2) = (k - 1)(- (k + 1) / 2).
y - (7 / 2) = (1 - k)(k + 1) / 2.
2y - 7 = (1 - k)(k + 1).
2y - 7 = k + 1 - k^2 - k.
2y - 7 = -k^2 - 2k + 1.
k^2 + 2k + 2y - 8 = 0.
So, the equation of the perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has a y-intercept of -8.
The x-coordinate of the midpoint is (1 + k) / 2 = (k + 1) / 2.
The y-coordinate of the midpoint is (4 + 3) / 2 = 7 / 2.
So, the midpoint is M((k + 1) / 2, 7 / 2).
The slope of the line segment joining P and Q is (3 - 4) / (k - 1) = -1 / (k - 1).
Since the perpendicular bisector is perpendicular to the line segment, its slope will be the negative reciprocal of the slope of the line segment. So, the slope of the perpendicular bisector is (k - 1) / 1.
Since the perpendicular bisector passes through the midpoint M, we can use the point-slope form of a line to find the equation of the perpendicular bisector.
y - (7 / 2) = (k - 1)(x - (k + 1) / 2).
To find the y-intercept, we can set x = 0 and solve for y:
y - (7 / 2) = (k - 1)(0 - (k + 1) / 2).
y - (7 / 2) = (k - 1)(- (k + 1) / 2).
y - (7 / 2) = (1 - k)(k + 1) / 2.
2y - 7 = (1 - k)(k + 1).
2y - 7 = k + 1 - k^2 - k.
2y - 7 = -k^2 - 2k + 1.
k^2 + 2k + 2y - 8 = 0.
So, the equation of the perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has a y-intercept of -8.
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The perpendicular bisector of the line segment joining P (1, 4) and Q(k, 3) has y-intercept –4. Then a possible value of k is[2008]a)1b)2c)–2d)– 4Correct answer is option 'D'. Can you explain this answer?
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