Coordinate Geometry - Study Mode

[#1] The graph of the equation x - 7y = -72, intersects the y-axis at P(α, β) and the graph of 6x + y - 15 = 0, intersects the x-axis at Q(γ, δ). What is the value of α + β + γ + δ?
Correct Answer

(D) $$frac{{17}}{2}$$

Explanation

Solution: At y-axis, x = 0 0 - 7y = -42 y = 6 (α, β) = (0, 6) At x-axis, y = 0 6x + 0 = 15 x = $$frac{5}{2}$$ (γ, δ) = $$left( {frac{5}{2},,0}
ight)$$ α + β + γ + δ = 0 + 6 + $$frac{5}{2}$$ + 0 = $$frac{{17}}{2}$$

[#2] What is the slope of the line parallel to the line passing through the points (5, -1) and (4, -4)?
Correct Answer

(C) 3

Explanation

Solution: $$eqalign{
& {m_1}left( {{ ext{slope}}}
ight) = frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} cr
& Rightarrow frac{{ - 4 - left( { - 1}
ight)}}{{4 - 5}} cr
& Rightarrow frac{{ - 3}}{{ - 1}} = left( 3
ight) cr
& herefore {m_1} = {m_2}left( {{ ext{In parallel condition}}}
ight) cr
& herefore {m_2} = 3 cr} $$

[#3] Find the area of quadrilateral formed by joining points (4, 2), (8, 2), (8, 14) and (4, 10).
Correct Answer

(D) 40 sq. units

Explanation

Solution: $$eqalign{
& { ext{Area of }}Delta ABC cr
& = frac{1}{2}left[ {{x_1}left( {{y_2} - {y_3}}
ight) + {x_2}left( {{y_3} - {y_1}}
ight) + {x_3}left( {{y_1} - {y_2}}
ight)}
ight] cr
& = frac{1}{2}left[ {4left( {2 - 14}
ight) + 8left( {14 - 2}
ight) + 8left( {2 - 2}
ight)}
ight] cr
& = frac{1}{2}left[ {4 imes left( { - 12}
ight) + 8left( {12}
ight) + 8left( 0
ight)}
ight] cr
& = frac{1}{2}left[ { - 48 + 96}
ight] cr
& = frac{1}{2} imes 48 cr
& = 24{ ext{ sq}}{ ext{. units}} cr
& { ext{Area of }}Delta ACD cr
& = frac{1}{2}left[ {{x_1}left( {{y_2} - {y_3}}
ight) + {x_2}left( {{y_3} - {y_1}}
ight) + {x_3}left( {{y_1} - {y_2}}
ight)}
ight] cr
& = frac{1}{2}left[ {4left( {14 - 10}
ight) + 8left( {10 - 2}
ight) + 4left( {2 - 14}
ight)}
ight] cr
& = frac{1}{2}left[ {4 imes 4 + 8 imes 8 + 4 imes left( { - 12}
ight)}
ight] cr
& = frac{1}{2}left[ {16 + 64 - 48}
ight] cr
& = frac{1}{2} imes 32 cr
& = 16{ ext{ sq}}{ ext{. units}} cr
& { ext{Hence Area of Quadrilateral }}ABCD cr
& = { ext{Area of }}left( {Delta ABC + Delta ACD}
ight) cr
& = 24 + 16 cr
& = 40{ ext{ sq}}{ ext{. units}} cr} $$

[#4] What is the slope of the line, parallel to the line 3x - 6y = 4?
Correct Answer

(B) $$frac{1}{2}$$

Explanation

Solution: $$eqalign{
& 3x - 6y = 4 cr
& Rightarrow 6y = 3x - 4 cr
& Rightarrow y = frac{{3x}}{6} - frac{4}{6} cr
& Rightarrow y = frac{1}{2}x - frac{2}{3} cr
& y = {m_1}x + c cr
& { ext{If lines are parallel then their slopes are equal}} cr
& herefore {m_1} = {m_2} = frac{1}{2} cr} $$

[#5] If the ordinate and abscissa of the point (k, 2k - 1) be equal, then the value of k is:
Correct Answer

(C) 1

Explanation

Solution: Since ordinate = abscissa ∴ k = 2k - 1 k = 1