Advertisements
Advertisements
Question
Find the equation of a line which is perpendicular to the line \[\sqrt{3}x - y + 5 = 0\] and which cuts off an intercept of 4 units with the negative direction of y-axis.
Solution
The line perpendicular to \[\sqrt{3}x - y + 5 = 0\] is \[x + \sqrt{3}y + \lambda = 0\].
It is given that the line \[x + \sqrt{3}y + \lambda = 0\] cuts off an intercept of 4 units with the negative direction of the y-axis.
This means that the line passes through \[\left( 0, - 4 \right)\].
\[\therefore 0 - \sqrt{3} \times 4 + \lambda = 0\]
\[ \Rightarrow \lambda = 4\sqrt{3}\]
Substituting the value of \[\lambda\], we get
\[x + \sqrt{3}y + 4\sqrt{3} = 0\], which is the equation of the required line.
APPEARS IN
RELATED QUESTIONS
Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having x intercept 3.
The line through the points (h, 3) and (4, 1) intersects the line 7x – 9y – 19 = 0. at right angle. Find the value of h.
Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes.
In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?
The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (−4, 1). Find the equation of the legs (perpendicular sides) of the triangle that are parallel to the axes.
Find the equation of a line which makes an angle of tan−1 (3) with the x-axis and cuts off an intercept of 4 units on negative direction of y-axis.
Find the equation of a line that has y-intercept −4 and is parallel to the line joining (2, −5) and (1, 2).
Find the equation of the right bisector of the line segment joining the points (3, 4) and (−1, 2).
Find the equation of the right bisector of the line segment joining the points A (1, 0) and B (2, 3).
Find the equations of the sides of the triangles the coordinates of whose angular point is respectively (1, 4), (2, −3) and (−1, −2).
For what values of a and b the intercepts cut off on the coordinate axes by the line ax + by + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2x − 3y + 6 = 0 on the axes.
Find the equation of a line for p = 5, α = 60°.
Find the equation of the line on which the length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is 30°.
If the straight line through the point P (3, 4) makes an angle π/6 with the x-axis and meets the line 12x + 5y + 10 = 0 at Q, find the length PQ.
Reduce the following equation to the normal form and find p and α in \[x + y + \sqrt{2} = 0\].
Reduce the following equation to the normal form and find p and α in x − 3 = 0.
Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line \[\sqrt{3}x + y + 2 = 0\].
Reduce the equation 3x − 2y + 6 = 0 to the intercept form and find the x and y intercepts.
Find the area of the triangle formed by the line y = m1 x + c1, y = m2 x + c2 and x = 0.
Find the area of the triangle formed by the line x + y − 6 = 0, x − 3y − 2 = 0 and 5x − 3y + 2 = 0.
Prove that the lines \[y = \sqrt{3}x + 1, y = 4 \text { and } y = - \sqrt{3}x + 2\] form an equilateral triangle.
Find the coordinates of the incentre and centroid of the triangle whose sides have the equations 3x− 4y = 0, 12y + 5x = 0 and y − 15 = 0.
Find the conditions that the straight lines y = m1 x + c1, y = m2 x + c2 and y = m3 x + c3 may meet in a point.
If the three lines ax + a2y + 1 = 0, bx + b2y + 1 = 0 and cx + c2y + 1 = 0 are concurrent, show that at least two of three constants a, b, c are equal.
Find the equation of the perpendicular bisector of the line joining the points (1, 3) and (3, 1).
Find the image of the point (2, 1) with respect to the line mirror x + y − 5 = 0.
Find the values of α so that the point P (α2, α) lies inside or on the triangle formed by the lines x − 5y+ 6 = 0, x − 3y + 2 = 0 and x − 2y − 3 = 0.
Find the values of the parameter a so that the point (a, 2) is an interior point of the triangle formed by the lines x + y − 4 = 0, 3x − 7y − 8 = 0 and 4x − y − 31 = 0.
The point which divides the join of (1, 2) and (3, 4) externally in the ratio 1 : 1
The number of real values of λ for which the lines x − 2y + 3 = 0, λx + 3y + 1 = 0 and 4x − λy + 2 = 0 are concurrent is
The equations of the sides AB, BC and CA of ∆ ABC are y − x = 2, x + 2y = 1 and 3x + y + 5 = 0 respectively. The equation of the altitude through B is
The centroid of a triangle is (2, 7) and two of its vertices are (4, 8) and (−2, 6). The third vertex is
Prove that every straight line has an equation of the form Ax + By + C = 0, where A, B and C are constants.
The inclination of the line x – y + 3 = 0 with the positive direction of x-axis is ______.
If the line `x/"a" + y/"b"` = 1 passes through the points (2, –3) and (4, –5), then (a, b) is ______.
A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is ______.
Reduce the following equation into slope-intercept form and find their slopes and the y-intercepts.
x + 7y = 0
Reduce the following equation into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
x − y = 4
