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Apr 2808:00 AM PDT
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16 Nov 2022, 11:56
Kudos
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D
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Difficulty:
35%
(medium)
Question Stats:
79% (03:00) correct
21%
(02:31)
wrong
based on 33
sessions
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In the xy-plane shown, the circle is centered at the origin. Line l passes through the origin and the point (3, 4). Line m is perpendicular to the circle at the point (3, 4). What is the the x-intercept of line m?
A. 6
B. 7
C. 8
D. 25/3
E. 9
Attachment:
Untitled.png [ 32.98 KiB | Viewed 1865 times ]
16 Nov 2022, 13:15
Kudos
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The product of slopes of two lines which are perpendicular to each other = -1
Let's assume that the point of tangency is Z.
Slope of OZ * Slope of m = -1
Slope of OZ = 4/3
Therefore slope of m = -3/4
Point (3,4) lies on line m. We have a point on line m, and we know the slope of line m. Hence, the equation of line m can be formulated. Once the equation is known, the x intercept can be found by replacing y = 0.
Option D
The detailed working is attached.
Let's assume that the point of tangency is Z.
Slope of OZ * Slope of m = -1
Slope of OZ = 4/3
Therefore slope of m = -3/4
Point (3,4) lies on line m. We have a point on line m, and we know the slope of line m. Hence, the equation of line m can be formulated. Once the equation is known, the x intercept can be found by replacing y = 0.
Option D
The detailed working is attached.
Attachments
Screenshot 2022-11-17 014110.png [ 151.68 KiB | Viewed 1693 times ]
16 Nov 2022, 14:21
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Bunuel
Line l passes through the origin and the point (3, 4), therefore the slope of line l is (4 - 0)/(3 - 0) = 4/3.
Since line m is perpendicular to the circle, it is perpendicular to line l as well. Thus, the slopes of line l and line m are negative reciprocals of each other, which means that slope of line m is -1/(4/3) = -3/4.
Since the slope of line m is -3/4, it's equation is of the form y = (-3/4)x + b. Since line m passes through (3, 4), substituting x = 3 and y = 4 should satisfy this equation. We can find the value of b as follows:
4 = (-3/4)(3) + b
4 = -9/4 + b
b = 4 + 9/4 = 25/4
Thus, the equation of line m is y = (-3/4)x + 25/4. To find the x-intercept of line m, we simply substitute y = 0:
0 = (-3/4)x + 25/4
-25/4 = (-3/4)x
x = 25/3
Answer: D
