Coordinate Geometry - Study Mode

[#31] The graph of the equations 25x + 75y = 225 and x = 9 meet at the point:
Correct Answer

(B) (9, 0)

Explanation

Solution: 25x + 75y = 225 . . . . . . (i) x = 9 . . . . . . (ii) For meeting point, solving the equation (i) & (ii) ⇒ 25 × 9 + 75y = 225 ⇒ 75y = 225 - 225 ⇒ y = 0 ∴ Meeting point = (9, 0)

[#32] Slope of the side DA of the rectangle ABCD is $$frac{5}{3}$$. What is the slope of the side AB?
Correct Answer

(D) $$frac{{ - 3}}{5}$$

Explanation

Solution: In given rectangle Let $$square $$ ABCD Angle of rectangle = 90° AB ⊥ DA ∴ Slope $$left[ {{{ ext{m}}_1} = frac{{ - 1}}{{{{ ext{m}}_2}}}}
ight]$$ xa0 (when lines are perpendicular to each other) Slope of side DA (m 1 ) = $$frac{5}{3}$$ Slope of side AB (m 2 ) = $$frac{{ - 3}}{5}$$

[#33] For triangle ABC, find equation of median AD if co-ordinates of points A, B and C are (2, -4), (3, 0) and (5, -2) respectively?
Correct Answer

(A) 3x - 2y = 14

Explanation

Solution: Co-ordinates of mid point (D) which lies on the line BC = $$left( {frac{{3 + 5}}{2},,frac{{0 - 2}}{2}}
ight) = left( {4,, - 1}
ight)$$ Now, Co-ordinates of line AD = A(2, -4), D(4, -1) ∴ Equation of the line which passes through the two point (x 1 , y 1 ), & (x 2 , y 2 ) ⇒ y - y 1 = m(x - x 1 ) ∴ required equation of the line = y + 4 = m(x - 2) $$eqalign{
& left[ {x08ecause m = frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}}
ight] cr
& herefore m = frac{{ - 1 + 4}}{{4 - 2}} = frac{3}{2} cr
& Rightarrow y + 4 = frac{3}{2}left[ {x - 2}
ight] cr
& Rightarrow 2y + 8 = 3x - 6 cr
& Rightarrow 3x - 2y = 14 cr} $$

[#34] The length of the intercept of the graph of the equation 9x - 12y = 108 between the two axes is-
Correct Answer

(A) 15 units

Explanation

Solution: $$eqalign{
& 9x - 12y = 108 cr
& Rightarrow frac{{9x}}{{108}} - frac{{12y}}{{108}} = 1 cr
& Rightarrow frac{x}{{12}} - frac{y}{9} = 1 cr} $$ $$eqalign{
& { ext{Length of Intercept}} = sqrt {{{12}^2} + {9^2}} cr
& = sqrt {144 + 81} cr
& = sqrt {225} cr
& = 15{ ext{ units}} cr} $$

[#35] What is the equation of the line whose y-intercept is $$ - frac{3}{4}$$xa0and making an angle of 45° with the positive x-axis?
Correct Answer

(A) 4x - 4y = 3

Explanation

Solution: Standard equation of the line y = mx + c ∴ m = tanθ = tan45° (θ = 45°) Given m = 1 ∴ c = $$ - frac{3}{4}$$ New required equation of the line ⇒ y = mx + c ⇒ y = 1 × x $$ - frac{3}{4}$$ ⇒ 4y = 4x - 3 ⇒ 4x - 4y = 3