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Zorluk: ZorLinear Functions and Graphs

In the xyxy-plane, the graphs of two linear functions, ff and gg, are perpendicular lines that intersect at the point (12,k)(12, k), where kk is a positive constant. If the yy-intercept of the graph of ff is (0,24)(0, 24) and the yy-intercept of the graph of gg is (0,−6)(0, -6), what is the value of kk?

Cevap: 18

Cevap

18
To find the value of kk, we determine the slopes of the two lines. The line representing ff passes through (0,24)(0, 24) and (12,k)(12, k), so its slope is mf=k−2412m_f = \frac{k - 24}{12}. The line representing gg passes through (0,−6)(0, -6) and (12,k)(12, k), so its slope is mg=k+612m_g = \frac{k + 6}{12}. Since the two lines are perpendicular, the product of their slopes is −1-1. This gives the equation (k−2412)(k+612)=−1\left(\frac{k - 24}{12}\right)\left(\frac{k + 6}{12}\right) = -1. Multiplying both sides by 144144 yields (k−24)(k+6)=−144(k - 24)(k + 6) = -144. Expanding the left side gives k2−18k−144=−144k^2 - 18k - 144 = -144. Adding 144144 to both sides results in k2−18k=0k^2 - 18k = 0, which factors as k(k−18)=0k(k - 18) = 0. Since kk is a positive constant, kk must be 1818.

Adım Adım Çözüm

1
Determine the slopes of the lines ff and gg in terms of kk.
The slope of ff is mf=k−2412−0=k−2412m_f = \frac{k - 24}{12 - 0} = \frac{k - 24}{12}. The slope of gg is mg=k−(−6)12−0=k+612m_g = \frac{k - (-6)}{12 - 0} = \frac{k + 6}{12}.
We use the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} for each line passing through their respective yy-intercepts and their intersection point (12,k)(12, k).
2
Set up an equation using the perpendicular condition.
mf⋅mg=−1  ⟹  (k−2412)(k+612)=−1  ⟹  (k−24)(k+6)144=−1m_f \cdot m_g = -1 \implies \left(\frac{k - 24}{12}\right)\left(\frac{k + 6}{12}\right) = -1 \implies \frac{(k - 24)(k + 6)}{144} = -1
Since the graphs of ff and gg are perpendicular, the product of their slopes must equal −1-1.
3
Solve the quadratic equation for kk.
(k−24)(k+6)=−144  ⟹  k2−18k−144=−144  ⟹  k2−18k=0  ⟹  k(k−18)=0(k - 24)(k + 6) = -144 \implies k^2 - 18k - 144 = -144 \implies k^2 - 18k = 0 \implies k(k - 18) = 0. Since kk must be positive, k=18k = 18.
Multiplying by 144144, expanding, and factoring the quadratic equation gives the possible values k=0k = 0 and k=18k = 18. The problem states kk is positive, so k=18k = 18.

Anahtar Kavram

Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is −1-1.
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Linear Functions and Graphs

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