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Difficulty: Very hardCoordinate Geometry and Lines

In the xyxy-plane, line ℓ1\ell_1 passes through the point (3,7)(3, 7) and is perpendicular to line ℓ2\ell_2, which is defined by the equation 2x−5y=102x - 5y = 10. Line ℓ3\ell_3 is parallel to line ℓ1\ell_1 and has a yy-intercept that is 66 units greater than the yy-intercept of line ℓ1\ell_1. If line ℓ3\ell_3 intersects the xx-axis at the point (a,0)(a, 0), what is the value of aa?

Answer: 8.2

Answer

The value of aa is 8.2 (or 415\frac{41}{5}).
Converting 2x−5y=102x - 5y = 10 to slope-intercept form yields y=25x−2y = \frac{2}{5}x - 2, so the slope of line ℓ2\ell_2 is 25\frac{2}{5}. Line ℓ1\ell_1 is perpendicular to ℓ2\ell_2, giving it a slope of m1=−52m_1 = -\frac{5}{2}. Using the point (3,7)(3, 7), the line equation for ℓ1\ell_1 is y−7=−2.5(x−3)y - 7 = -2.5(x - 3), which simplifies to y=−2.5x+14.5y = -2.5x + 14.5. Line ℓ3\ell_3 is parallel to ℓ1\ell_1, so m3=−2.5m_3 = -2.5, and its yy-intercept is 14.5+6=20.514.5 + 6 = 20.5. Writing the equation for line ℓ3\ell_3 as y=−2.5x+20.5y = -2.5x + 20.5 and setting y=0y = 0 gives 0=−2.5a+20.5  ⟹  2.5a=20.5  ⟹  a=8.20 = -2.5a + 20.5 \implies 2.5a = 20.5 \implies a = 8.2.

Step-by-Step Solution

1
Determine the slope of line ℓ2\ell_2
The slope of line ℓ2\ell_2 is 25\frac{2}{5}.
Convert 2x−5y=102x - 5y = 10 into slope-intercept form y=25x−2y = \frac{2}{5}x - 2.
2
Determine the slope of line ℓ1\ell_1
The slope of line ℓ1\ell_1 is −52-\frac{5}{2}.
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Find the yy-intercept of line ℓ1\ell_1
The yy-intercept of ℓ1\ell_1 is 292=14.5\frac{29}{2} = 14.5.
Apply point-slope form with point (3,7)(3, 7): y−7=−52(x−3)  ⟹  y=−52x+292y - 7 = -\frac{5}{2}(x - 3) \implies y = -\frac{5}{2}x + \frac{29}{2}.
4
Construct the equation for line ℓ3\ell_3
The equation of ℓ3\ell_3 is y=−52x+412y = -\frac{5}{2}x + \frac{41}{2}.
Line ℓ3\ell_3 has the same slope as ℓ1\ell_1 (−52-\frac{5}{2}) and its yy-intercept is 14.5+6=20.5=41214.5 + 6 = 20.5 = \frac{41}{2}.
5
Calculate the xx-intercept coordinate aa of line ℓ3\ell_3
a=8.2a = 8.2.
Substitute y=0y = 0 into the equation for ℓ3\ell_3: 0=−52a+412  ⟹  5a=41  ⟹  a=8.20 = -\frac{5}{2}a + \frac{41}{2} \implies 5a = 41 \implies a = 8.2.

Key Concept

Perpendicular and parallel slopes, point-slope equation derivation, line transformations, and intercept determination.
Estimated Time:2m 30s
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Topics

Coordinate Geometry and Lines

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