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Straight Lines

Multiple Choice Questions

81.

The equation of straight line through the intersection of the lines x - 2y = 1 and x + 3y = 2 and parallel to 3x + 4y = 0 is

3x + 4y + 5 = 0
3x + 4y - 10 = 0
3x + 4y - 5 = 0
3x + 4y + 6 = 0

82.

To the lines ax² + 2hxy + by² = 0, the lines a²x² + 2h(a + b) xy + b²y² = 0 are

equally inclined
perpendicular
bisector of the angle
None of these

83.

A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. What is its y intercept

84.

The equation of straight line through the intersection of line 2x +y = 1 and 3x + 2y = 5 and passing through the origin is

7x + 3y = 0
7x - y = 0
3x + 2y = 0
x + y = 0

85.

The condition that the line lx + my = 1 may be normal to the curve y² = 4ax, is

al³ - 2alm² = m²
al² + 2alm³ = m²
al³ + 2alm² = m³
al³ + 2alm² = m²

86.
If the lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{2} = \frac{y - k}{3} = \frac{z}{1}$ intersect, then the value of k, is

$\frac{3}{2}$

$\frac{9}{2}$

$- \frac{2}{9}$

- $\frac{3}{2}$

$\frac{9}{2}$

We have the lines

$\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ ...(i)

$\frac{x - 3}{2} = \frac{y - k}{3} = \frac{z}{1}$ ...(ii)

Let a point (2r + 1, 3r - 1, 4r + 1) be on the line Eq. (i). If this is an intersection point of both the lines, then it will lie on Eq. (ii), also

$∴ \frac{2 r + 1 - 3}{1} = \frac{3 r - 1 - k}{2} = \frac{4 r + 1}{1}$ ...(iii)

Taking first and third part of Eq. (iii), we get

2r - 2 = 4r + 1

$\Rightarrow r = - \frac{3}{2}$

Taking second and third part of Eq. (iii), we get

3r - 1 - k = 8r + 2

$\Rightarrow$ 3r - 1 - k - 8r - 2 = 0

$\Rightarrow k = - 5 r - 3 \Rightarrow k = - 5 (- \frac{3}{2}) - 3 (∵ r = - \frac{3}{2}) \Rightarrow k = \frac{15}{2} - 3 \Rightarrow k = \frac{9}{2}$

87.

The area (in square unit) of the triangle formed by x + y + 1 = 0 and the pair of straight lines x² - 3xy + 2y² = 0 is

$\frac{7}{12}$
$\frac{5}{12}$
$\frac{1}{12}$
$\frac{1}{6}$

88.

The pairs of straight lines x² - 3xy + 2y²= 0 and x² - 3xy + 2y² + x - 2 = 0 form a

square but not rhombus
rhombus
parallelogram
rectangle but not a square

89.

The shortest distance between the straight lines through the points A₁ = (6, 2, 2) and A₂ = (- 4, 0, - 1) in the directions of (1, - 2, 2) and (3, - 2, - 2) is