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Zorluk: Çok zorParallel and Perpendicular Lines

In the standard (x,y)(x, y) coordinate plane, the perpendicular bisector of the line segment with endpoints (1,2)(1, 2) and (5,10)(5, 10) intersects the curve y=x2y = x^2 at two points. If one of these intersection points lies in the second quadrant, what is the yy-coordinate of this point?

Cevap: 9

Cevap

The correct answer is 99.
The perpendicular bisector of the segment connecting (1,2)(1, 2) and (5,10)(5, 10) passes through their midpoint (3,6)(3, 6) and has a slope of −12-\frac{1}{2} (the negative reciprocal of 22). The equation of this line is x+2y=15x + 2y = 15. Substituting y=x2y = x^2 yields the quadratic equation 2x2+x−15=02x^2 + x - 15 = 0, which factors into (2x−5)(x+3)=0(2x - 5)(x + 3) = 0. The solution in the second quadrant corresponds to the negative xx-coordinate, x=−3x = -3. Squaring this value gives a yy-coordinate of 99.

Adım Adım Çözüm

1
Find the midpoint of the segment with endpoints (1,2)(1, 2) and (5,10)(5, 10).
The midpoint is (3,6)(3, 6).
The perpendicular bisector of a segment passes through its midpoint.
2
Calculate the slope of the segment and the slope of the perpendicular bisector.
The slope of the segment is 22, and the perpendicular slope is −12-\frac{1}{2}.
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Write the equation of the perpendicular bisector.
The equation of the line is x+2y=15x + 2y = 15.
Using the point-slope form with point (3,6)(3, 6) and slope −12-\frac{1}{2} yields y−6=−12(x−3)y - 6 = -\frac{1}{2}(x - 3), which simplifies to x+2y=15x + 2y = 15.
4
Find the intersection points of the line and the curve y=x2y = x^2.
The xx-coordinates of the intersection points are 2.52.5 and −3-3.
Substituting y=x2y = x^2 into x+2y=15x + 2y = 15 gives the quadratic equation 2x2+x−15=02x^2 + x - 15 = 0, which factors as (2x−5)(x+3)=0(2x - 5)(x + 3) = 0.
5
Identify the point in the second quadrant and find its yy-coordinate.
The point is (−3,9)(-3, 9), so the yy-coordinate is 99.
A point in the second quadrant must have a negative xx-coordinate (x=−3x = -3) and a positive yy-coordinate (y=9y = 9).

Anahtar Kavram

Perpendicular bisectors and systems of linear-quadratic equations in coordinate geometry
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Konular

Parallel and Perpendicular Lines

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