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Difficulty: HardLinear Equations and Graphing

In a coordinate plane, two perpendicular lines, L1L_1 and L2L_2, intersect at a point on the positive yy-axis. The line L1L_1 passes through the point (−4,3)(-4, 3), and the line L2L_2 has an xx-intercept at (2.5,0)(2.5, 0). What is the yy-coordinate of the intersection point of L1L_1 and L2L_2?

Answer: 5

Answer

The y-coordinate of the intersection point is 5.
The intersection point on the positive yy-axis has coordinates (0,5)(0, 5). The slope of the line passing through (−4,3)(-4, 3) and (0,5)(0, 5) is 0.50.5. The slope of the line passing through (2.5,0)(2.5, 0) and (0,5)(0, 5) is −2-2. Since the product of these slopes is 0.5×(−2)=−10.5 \times (-2) = -1, the lines are perpendicular.

Step-by-Step Solution

1
Identify the coordinates of the intersection point.
The intersection point of L1L_1 and L2L_2 is (0,b)(0, b), where b>0b > 0.
Since the intersection point lies on the positive yy-axis, its xx-coordinate is 00 and its yy-coordinate bb must be positive.
2
Calculate the slope of L1L_1.
The slope of L1L_1 is m1=b−30−(−4)=b−34m_1 = \frac{b - 3}{0 - (-4)} = \frac{b - 3}{4}.
The slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} is applied to the points (−4,3)(-4, 3) and (0,b)(0, b).
3
Calculate the slope of L2L_2.
The slope of L2L_2 is m2=b−00−2.5=−b2.5m_2 = \frac{b - 0}{0 - 2.5} = -\frac{b}{2.5}.
The line L2L_2 passes through the yy-intercept (0,b)(0, b) and its xx-intercept (2.5,0)(2.5, 0).
4
Apply the perpendicular slope condition.
m1⋅m2=−1  ⟹  (b−34)(−b2.5)=−1  ⟹  b(b−3)=10m_1 \cdot m_2 = -1 \implies \left(\frac{b - 3}{4}\right)\left(-\frac{b}{2.5}\right) = -1 \implies b(b - 3) = 10.
Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is −1-1.
5
Solve the quadratic equation for bb.
Expanding the equation gives b2−3b−10=0b^2 - 3b - 10 = 0, which factors into (b−5)(b+2)=0(b - 5)(b + 2) = 0. Since b>0b > 0, the only valid solution is b=5b = 5.
Solving the factored equation yields b=5b = 5 or b=−2b = -2. The constraint that the intersection lies on the positive yy-axis rules out the negative value.

Key Concept

Perpendicular line slopes and coordinate intercepts
Estimated Time:2m 30s
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Topics

Linear Equations and Graphing

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