Advertisements
Advertisements
Question
The equations of the lines which pass through the point (3, –2) and are inclined at 60° to the line `sqrt(3) x + y` = 1 is ______.
Options
y + 2 = 0, `sqrt(3) x - y - 2 - 3 sqrt(3)` = 0
x – 2 = 0, `sqrt(3)x - y + 2 + 3 sqrt(3)` = 0
`sqrt(3) x - y - 2 - 3sqrt(3)` = 0
None of these
Advertisements
Solution
The equations of the lines which pass through the point (3, –2) and are inclined at 60° to the line `sqrt(3) x + y` = 1 is y + 2 = 0, `sqrt(3) x - y - 2 - 3 sqrt(3)` = 0.
Explanation:
Equation of line is given by `sqrt(3)x + y + 1` = 0
`sqrt(3)x + y = 1` = 0
⇒ y = `- sqrt(3)x - 1`
∴ Slope of this line, m1 `- sqrt(3)`
Let m2 be the slope of the required line
∴ tan θ = `|(m_1 - m_2)/(1 + m_1m_2)|`
⇒ tan 60° = `|(-sqrt(3) - m_2)/(1 + (- sqrt(3))m_2)|`
⇒ `sqrt(3) = +- ((- sqrt(3) - m_2)/(1 - sqrt(3)m_2))`
⇒ `sqrt(3) = (- sqrt(3) - m_2)/(1 - sqrt(3)m_2)` ....[Taking (+) sign]
⇒ `sqrt(3) - 3m_2 = - sqrt(3) -m_2`
⇒ `2m_2 = 2sqrt(3)`
⇒ m2 = `sqrt(3)`
And `sqrt(3) - ((- sqrt(3) - m_2)/(1 - sqrt(3)m_2))` ....[Taking (–) sign]
⇒ `sqrt(3) = (sqrt(3) + m_2)/(1 - sqrt(3)m_2)`
⇒ `sqrt(3) - 3m_2 = sqrt(3) + m_2`
⇒ 4m2 = 0
⇒ m2 = 0
∴ Equation of line passing through (3, – 2) with slope `sqrt(3)` is
y + 2 = `sqrt(3)(x - 3)`
⇒ y + 2 = `sqrt(3)x - 3sqrt(3)`
⇒ `sqrt(3)x - y - 2 - 3sqrt(3)` = 0
And the equation of line passing through (3, –2) with slope 0 is
y + 2 = 0(x – 3)
⇒ y + 2 = 0
APPEARS IN
RELATED QUESTIONS
Find the equation of the line parallel to x-axis and passing through (3, −5).
Find the equation of the line perpendicular to x-axis and having intercept − 2 on x-axis.
Find the equation of the line parallel to x-axis and having intercept − 2 on y-axis.
Find the equation of the line passing through \[(2, 2\sqrt{3})\] and inclined with x-axis at an angle of 75°.
Find the equation of the straight line passing through (3, −2) and making an angle of 60° with the positive direction of y-axis.
Find the equation of the line passing through the point (−3, 5) and perpendicular to the line joining (2, 5) and (−3, 6).
Find the equation of the straight lines passing through the following pair of point :
(0, 0) and (2, −2)
Find the equation of the straight lines passing through the following pair of point :
(a cos α, a sin α) and (a cos β, a sin β)
Find the equations of the sides of the triangles the coordinates of whose angular point is respectively (0, 1), (2, 0) and (−1, −2).
Find the equations of the medians of a triangle, the coordinates of whose vertices are (−1, 6), (−3, −9) and (5, −8).
Find the equation to the straight line cutting off intercepts 3 and 2 from the axes.
Find the equation of the straight line which passes through (1, −2) and cuts off equal intercepts on the axes.
Find the equation to the straight line which passes through the point (5, 6) and has intercepts on the axes
(i) equal in magnitude and both positive,
(ii) equal in magnitude but opposite in sign.
Find the equation of the line which passes through the point (− 4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5 : 3 by this point.
Find the equation of a line which passes through the point (22, −6) and is such that the intercept of x-axis exceeds the intercept of y-axis by 5.
Find the equation of the line, which passes through P (1, −7) and meets the axes at A and Brespectively so that 4 AP − 3 BP = 0.
Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.
Find the equations of the straight lines each of which passes through the point (3, 2) and cuts off intercepts a and b respectively on X and Y-axes such that a − b = 2.
A straight line drawn through the point A (2, 1) making an angle π/4 with positive x-axis intersects another line x + 2y + 1 = 0 in the point B. Find length AB.
If the straight line \[\frac{x}{a} + \frac{y}{b} = 1\] passes through the point of intersection of the lines x + y = 3 and 2x − 3y = 1 and is parallel to x − y − 6 = 0, find a and b.
Find the equation of a line passing through the point (2, 3) and parallel to the line 3x − 4y + 5 = 0.
Find the length of the perpendicular from the origin to the straight line joining the two points whose coordinates are (a cos α, a sin α) and (a cos β, a sin β).
Find the equation of the straight lines passing through the origin and making an angle of 45° with the straight line \[\sqrt{3}x + y = 11\].
Find the equations to the straight lines which pass through the origin and are inclined at an angle of 75° to the straight line \[x + y + \sqrt{3}\left( y - x \right) = a\].
Two sides of an isosceles triangle are given by the equations 7x − y + 3 = 0 and x + y − 3 = 0 and its third side passes through the point (1, −10). Determine the equation of the third side.
Show that the straight lines given by (2 + k) x + (1 + k) y = 5 + 7k for different values of k pass through a fixed point. Also, find that point.
Find the equation of the straight line passing through the point of intersection of 2x + y − 1 = 0 and x + 3y − 2 = 0 and making with the coordinate axes a triangle of area \[\frac{3}{8}\] sq. units.
If the diagonals of the quadrilateral formed by the lines l1x + m1y + n1 = 0, l2x + m2y + n2 = 0, l1x + m1y + n1' = 0 and l2x + m2y + n2' = 0 are perpendicular, then write the value of l12 − l22 + m12 − m22.
Write the integral values of m for which the x-coordinate of the point of intersection of the lines y = mx + 1 and 3x + 4y = 9 is an integer.
Write the equation of the line passing through the point (1, −2) and cutting off equal intercepts from the axes.
Find the equation of lines passing through (1, 2) and making angle 30° with y-axis.
In what direction should a line be drawn through the point (1, 2) so that its point of intersection with the line x + y = 4 is at a distance `sqrt(6)/3` from the given point.
The equation of the line joining the point (3, 5) to the point of intersection of the lines 4x + y – 1 = 0 and 7x – 3y – 35 = 0 is equidistant from the points (0, 0) and (8, 34).
