Advertisements
Advertisements
प्रश्न
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 ______.
विकल्प
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
उत्तर
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
संबंधित प्रश्न
Find the equation of the line parallel to x-axis and having intercept − 2 on y-axis.
Find the equation of the straight line passing through the point (6, 2) and having slope − 3.
Find the equation of the straight line passing through (3, −2) and making an angle of 60° with the positive direction of y-axis.
Prove that the perpendicular drawn from the point (4, 1) on the join of (2, −1) and (6, 5) divides it in the ratio 5 : 8.
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 which bisects the distance between the points (a, b), (a', b') and also bisects the distance between the points (−a, b) and (a', −b').
The vertices of a quadrilateral are A (−2, 6), B (1, 2), C (10, 4), and D (7, 8). Find the equation of its diagonals.
Find the equation to the straight line cutting off intercepts 3 and 2 from the axes.
Find the equation of the line which passes through the point (3, 4) and is such that the portion of it intercepted between the axes is divided by the point in the ratio 2:3.
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 equation of the straight line passing through the point (2, 1) and bisecting the portion of the straight line 3x − 5y = 15 lying between the axes.
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 equation of the straight line through the point (α, β) and perpendicular to the line lx + my + n = 0.
The line 2x + 3y = 12 meets the x-axis at A and y-axis at B. The line through (5, 5) perpendicular to AB meets the x-axis and the line AB at C and E respectively. If O is the origin of coordinates, find the area of figure OCEB.
Find the distance of the point (1, 2) from the straight line with slope 5 and passing through the point of intersection of x + 2y = 5 and x − 3y = 7.
The equation of one side of an equilateral triangle is x − y = 0 and one vertex is \[(2 + \sqrt{3}, 5)\]. Prove that a second side is \[y + (2 - \sqrt{3}) x = 6\] and find the equation of the third side.
Find the equations of two straight lines passing through (1, 2) and making an angle of 60° with the line x + y = 0. Find also the area of the triangle formed by the three lines.
Prove that the family of lines represented by x (1 + λ) + y (2 − λ) + 5 = 0, λ being arbitrary, pass through a fixed point. Also, find the fixed point.
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.
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 locus of the mid-points of the portion of the line x sinθ+ y cosθ = p intercepted between the axes.
The equation of the line passing through (1, 5) and perpendicular to the line 3x − 5y + 7 = 0 is
The inclination of the straight line passing through the point (−3, 6) and the mid-point of the line joining the point (4, −5) and (−2, 9) is
Find the equation of lines passing through (1, 2) and making angle 30° with y-axis.
Find the equation of the line passing through the point of intersection of 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.
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.
A straight line moves so that the sum of the reciprocals of its intercepts made on axes is constant. Show that the line passes through a fixed point.
Find the equations of the lines through the point of intersection of the lines x – y + 1 = 0 and 2x – 3y + 5 = 0 and whose distance from the point (3, 2) is `7/5`
The straight line 5x + 4y = 0 passes through the point of intersection of the straight lines x + 2y – 10 = 0 and 2x + y + 5 = 0.
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).
