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Number of integral values of 'k' for which the chord of the circle x2 + y2 = 125 passing through P(8, k) gets bisected at P (8, k) and has integral slope is
- a)8
- b)6
- c)4
- d)2
Correct answer is option 'B'. Can you explain this answer?
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Number of integral values of k for which the chord of the circle x2 + ...
The slope of the chord is
⇒ k = ± 1, ± 2, ± 4, ± 8 but (8, k) must also lie inside the circle x2 + y2 = 125
⇒ 64 + k2 – 125 < 0
⇒ k2 < 61
⇒ k can be equal to ± 1, ± 2, ± 4
⇒ 6 values
⇒ k = ± 1, ± 2, ± 4, ± 8 but (8, k) must also lie inside the circle x2 + y2 = 125
⇒ 64 + k2 – 125 < 0
⇒ k2 < 61
⇒ k can be equal to ± 1, ± 2, ± 4
⇒ 6 values
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Number of integral values of k for which the chord of the circle x2 + ...
To find the number of integral values of k for which the chord of the circle x^2 + y^2 = 125 passing through P(8, k) gets bisected at P(8, k) and has an integral slope, we need to analyze the given conditions.
Condition 1: The chord bisects at P(8, k)
If the chord bisects at P(8, k), it means that the midpoint of the chord lies at the point P(8, k). Let the coordinates of the other end of the chord be Q(x, y). Since the midpoint of a chord is the average of its endpoints, we have:
(x + 8)/2 = 8
x + 8 = 16
x = 8
Therefore, the other end of the chord is Q(8, y).
Condition 2: The chord has an integral slope
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1)/(x2 - x1)
In this case, the slope of the chord is given by:
m = (y - k)/(8 - 8)
m = (y - k)/0
Since the denominator is zero, the slope of the chord is undefined. This means that the chord is vertical and parallel to the y-axis.
Condition 3: The chord passes through the circle
Substituting the coordinates of Q(8, y) into the equation of the circle, we get:
8^2 + y^2 = 125
64 + y^2 = 125
y^2 = 125 - 64
y^2 = 61
Taking the square root of both sides, we get:
y = ±√61
Since the chord is vertical, the two possible values of k are √61 and -√61.
Number of integral values of k
Since k needs to be an integer, we can see that only one value (√61) satisfies this condition. Therefore, there is only one integral value of k for which the chord of the circle x^2 + y^2 = 125 passing through P(8, k) gets bisected at P(8, k) and has an integral slope.
Therefore, the correct answer is option 'B' (6 integral values of k).
Condition 1: The chord bisects at P(8, k)
If the chord bisects at P(8, k), it means that the midpoint of the chord lies at the point P(8, k). Let the coordinates of the other end of the chord be Q(x, y). Since the midpoint of a chord is the average of its endpoints, we have:
(x + 8)/2 = 8
x + 8 = 16
x = 8
Therefore, the other end of the chord is Q(8, y).
Condition 2: The chord has an integral slope
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1)/(x2 - x1)
In this case, the slope of the chord is given by:
m = (y - k)/(8 - 8)
m = (y - k)/0
Since the denominator is zero, the slope of the chord is undefined. This means that the chord is vertical and parallel to the y-axis.
Condition 3: The chord passes through the circle
Substituting the coordinates of Q(8, y) into the equation of the circle, we get:
8^2 + y^2 = 125
64 + y^2 = 125
y^2 = 125 - 64
y^2 = 61
Taking the square root of both sides, we get:
y = ±√61
Since the chord is vertical, the two possible values of k are √61 and -√61.
Number of integral values of k
Since k needs to be an integer, we can see that only one value (√61) satisfies this condition. Therefore, there is only one integral value of k for which the chord of the circle x^2 + y^2 = 125 passing through P(8, k) gets bisected at P(8, k) and has an integral slope.
Therefore, the correct answer is option 'B' (6 integral values of k).
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Number of integral values of k for which the chord of the circle x2 + y2 = 125 passing through P(8, k) gets bisected at P (8, k) and has integral slope isa)8b)6c)4d)2Correct answer is option 'B'. Can you explain this answer?
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