Coordinate Geometry - Study Mode

[#21] What would be the equation of the line, which intercepts x-axis at -5 and is perpendicular to the line y = 2x + 3?
Correct Answer

(C) x + 2y = -5

Explanation

Solution: Slope of line y = mx + c is m y = 2x + c, slope (m 1 ) = 2 Lines are perpendicular to each other $${m_1} = frac{{ - 1}}{{{m_2}}}$$ Slope of perpendicular line $$left( {{m_2}}
ight) = frac{{ - 1}}{2}$$ on x axis y = 0 Equation of line which passes through (-5, 0) (y - y 1 ) = slope (m 2 )(x - x 1 ) y - 0 = $$frac{{ - 1}}{2}$$ (x + 5) 2y = -x - 5 x + 2y = -5

[#22] The length of the portion of the straight line 3x + 4y = 12 intercepted between the axes is:
Correct Answer

(A) 5

Explanation

Solution: $$eqalign{
& 3x + 4y = 12 cr
& Rightarrow frac{{3x}}{{12}} + frac{{4y}}{{12}} = 1 cr
& Rightarrow frac{x}{4} + frac{y}{3} = 1 cr
& herefore { ext{Length of intercept AB}} = sqrt {{4^2} + {3^3}} cr
& = sqrt {25} cr
& = 5{ ext{ units}} cr} $$

[#23] P(4, 2) and R(-2, 0) are vertices of a rhombus PQRS. What is the equation of diagonal QS?
Correct Answer

(B) 3x + y = 4

Explanation

Solution: Slope of line PR $$eqalign{
& Rightarrow {m_1} = frac{{0 - 2}}{{ - 2 - 4}} cr
& Rightarrow {m_1} = frac{{ - 2}}{{ - 6}} cr
& Rightarrow {m_1} = frac{1}{3} cr} $$ ∴ Slope of line QS = -3 = m 2 {As it is perpendicular to PR} Coordinates of point O $$ Rightarrow left[ {frac{{ - 2 + 4}}{2},,frac{{2 + 0}}{2}}
ight] Rightarrow left( {1,,1}
ight)$$ ∴ Equation of line QS which passes through point O(1, 1) ⇒ y - 1 = m 2 (x - 1) ⇒ y - 1 = -3(x - 1) ⇒ y - 1 = -3x + 3 ⇒ 3x + y = 4

[#24] What is the equation of the line perpendicular to the line 2x + 3y = -6 and having Y-intercept 3?
Correct Answer

(B) 3x - 2y = -6

Explanation

Solution: $$eqalign{
& 2x + 3y = - 6 cr
& y = - frac{2}{3}x - frac{6}{3} cr
& herefore { ext{Slope}} = - frac{2}{3} cr} $$ ⇒ Two lines are perpendicular if m 1 × m 2 = -1 Y intercept = 3 ⇒ y = mx + c ⇒ m 2 = $$frac{3}{2}$$ The equation of line with slope $$frac{3}{2}$$ is:- y - y 1 = m 2 (x - x 1 ) where, (x 1 y 1 ) = (0, 3) y - 3 = $$frac{3}{2}$$(x - 0) 2y - 6 = 3x 3x - 2y = -6

[#25] The line passing through the point (5, a) and point (4, 3) is perpendicular to the line x - 6y = 8. What is the value of 'a'?
Correct Answer

(A) -3

Explanation

Solution: Given, Equation of line x - 6y = 8 we write it as $$y = frac{x}{6} - frac{8}{6}$$ ∴ y = mx (where m is a slope) ∴ m 1 = $$frac{1}{6}$$ If lines are perpendicular then product of their slopes is equal to -1 m 1 × m 2 = -1 ∴ $$frac{1}{6}$$ × m 2 = -1 m 2 = - 6 ∴ Slope of perpendicular line = - 6 Perpendicular line which passes through the points (5, a) and (4, 3) $$eqalign{
& herefore frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = {m_2} cr
& Rightarrow frac{{3 - a}}{{4 - 5}} = - 6 cr
& Rightarrow 3 - a = + 6 imes + 1 cr
& herefore a = - 3 cr} $$