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Straight Lines

Multiple Choice Questions

61.

What is the equation of the straight line which passes through the point of intersection of the straight lines x + 2y=5 and 3x + 7y = 17 and is perpendicular to the straight line 3x + 4y = 10 ?

4x + 3y + 2 = 0
4x - y + 2 = 0
4x - 3y - 2 = 0
4x - 3y + 2 = 0

62.

If (a, b) is at unit distance from the line 8x + 6y + 1 = 0, then which of the following conditions are correct ?

1. 3a - 4b - 4 = 0

2. 8a + 6b + 11 = 0

3. 8a + 6b - 9 = 0

Select the correct answer using the code given below :

1 and 2 only
2 and 3 only
1 and 3 only
1, 2 and 3

63.

A straight line cuts off an intercept of 2 units on the positive direction of x-axis and passes through the point ( - 3, 5). What is the foot of the perpendicular drawn from the point (3, 3) on this line

(1, 3)
(2, 0)
(0, 2)
(1, 1)

64.

If the point (x, y, - 3), (2, 0, - 1) and (4, 2, 3) lie on a straight line, then what are the values of x and y respectively ?

1, - 1
- 1, 1
0, 2
3, 4

65.

An equilateral triangle has one vertex at (- 1, - 1) and another vertex at $(- \sqrt{3}, \sqrt{3})$ . The third vertex may lie on

$(- \sqrt{2}, \sqrt{2})$
$(\sqrt{2}, - \sqrt{2})$
(1, 1)
(1, - 1)

66.

What is the distance between the points P(m.cos2α, m.sin2α) and Q(m.cos2β, m.sin2β) ?

$|2 msin (α - β)|$
$|2 mcos (α - β)|$
$|2 msin ((2 α - 2 β))|$
$|msin (2 α - 2 β)|$

67.
What is the angle between the lines xcos(α) + ysin(α) = α and xsin(β) – ycos(β) = α ?

$β - α$

$π + β - α$

$\frac{(π + 2 β + 2 α)}{2}$

$\frac{(π - 2 β + 2 α)}{2}$

$\frac{(π - 2 β + 2 α)}{2}$

The lines are,

$xcos (α) + ysin (α) = α and xsin (β) - ycos (β) = α For finding angle between lines firstly calculate slope, ∴ xcos (α) + ysin (α) = α \Rightarrow ysin (α) = α - xcos (α) y = - x (\frac{\cos (α)}{\sin (α)}) - \frac{α}{\sin (α)} Then slope m_{1} = - \frac{\cos (α)}{\sin (α)} ∴ xsin (β) - ycos (β) = α \Rightarrow ycos (β) = xsin (β) - α Then slope m_{2} = - \frac{\sin (β)}{\cos (β)} We know, the formula for calculating the angle between two lines, \tan (θ) = |\frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}| \tan (θ) = |\frac{- \frac{\cos (α)}{\sin (α)} - \frac{\sin (β)}{\cos (β)}}{1 + (- \frac{\cos (α)}{\sin (α)}) (\frac{\sin (β)}{\cos (β)})}| \Rightarrow \tan (θ) = |\frac{- (\cos (α) \cos (β) + \sin (α) \sin (β))}{\sin (α) \cos (β) - \cos (α) \sin (β)}| \Rightarrow \tan (θ) = |\frac{- \cos (α - β)}{\sin (α - β)}| \Rightarrow θ = \tan^{- 1} (|- (\cot (α - β))|) \Rightarrow θ = \tan^{- 1} (\tan (\frac{π}{2} + (α - β))) [∵ \tan (\frac{π}{2} + θ) = - \cot (θ)] \Rightarrow θ = \frac{π - 2 β + 2 α}{2}$

68.

The equation ax + by + c = 0 represents a straight line

for all real numbers a, b and c
only when a ≠ 0
only when b ≠ 0
only when at least one of a and b is non-zero

69.

A ladder 9 m long reaches a point 9 m below the top of a vertical flag staff. From the foot of the ladder, the elevation of the flag staff is 60^o. What is the height of the flag staff ?