Advertisements
Advertisements
Question
If a + b + c = 0, then the family of lines 3ax + by + 2c = 0 pass through fixed point
Options
(2, 2/3)
(2/3, 2)
(−2, 2/3)
none of these
Advertisements
Solution
(2/3, 2)
Given:
a + b + c = 0
Substituting c = − a − b in 3ax + by + 2c = 0, we get:
\[3ax + by - 2a - 2b = 0\]
\[ \Rightarrow a\left( 3x - 2 \right) + b\left( y - 2 \right) = 0\]
\[ \Rightarrow \left( 3x - 2 \right) + \frac{b}{a}\left( y - 2 \right) = 0\]
This line is of the form
\[L_1 + \lambda L_2 = 0\], which passes through the intersection of the lines \[L_1 \text { and } L_2\] i.e.
\[3x - 2 = 0 \text { and } y - 2 = 0\].
Solving \[3x - 2 = 0 \text { and } y - 2 = 0\],we get:
\[x = \frac{2}{3}, y = 2\]
Hence, the required fixed point is \[\left( \frac{2}{3}, 2 \right)\].
APPEARS IN
RELATED QUESTIONS
Find the equation of the line parallel to x-axis and passing through (3, −5).
Find the equations of the straight lines which pass through (4, 3) and are respectively parallel and perpendicular to the x-axis.
Find the equation of a line equidistant from the lines y = 10 and y = − 2.
Find the equation of the straight line passing through the point (6, 2) and having slope − 3.
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 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:
(a, b) and (a + c sin α, b + c cos α)
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 medians of a triangle, the coordinates of whose vertices are (−1, 6), (−3, −9) and (5, −8).
Find the equations to the diagonals of the rectangle the equations of whose sides are x = a, x = a', y= b and y = b'.
By using the concept of equation of a line, prove that the three points (−2, −2), (8, 2) and (3, 0) are collinear.
The length L (in centimeters) of a copper rod is a linear function of its celsius temperature C. In an experiment, if L = 124.942 when C = 20 and L = 125.134 when C = 110, express L in terms of C.
Find the equation to the straight line cutting off intercepts 3 and 2 from the axes.
Find the equation to the straight line which cuts off equal positive intercepts on the axes and their product is 25.
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 straight line which passes through the point (−3, 8) and cuts off positive intercepts on the coordinate axes whose sum is 7.
Find the equation of the straight line which passes through the point P (2, 6) and cuts the coordinate axes at the point A and B respectively so that \[\frac{AP}{BP} = \frac{2}{3}\] .
A line is such that its segment between the straight lines 5x − y − 4 = 0 and 3x + 4y − 4 = 0 is bisected at the point (1, 5). Obtain its equation.
Find the equation of straight line passing through (−2, −7) and having an intercept of length 3 between the straight lines 4x + 3y = 12 and 4x + 3y = 3.
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 the line passing through the intersection of the lines 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.
Find the equation of the straight line passing through the point of intersection of the lines 5x − 6y − 1 = 0 and 3x + 2y + 5 = 0 and perpendicular to the line 3x − 5y + 11 = 0 .
Find the equation of a line passing through the point (2, 3) and parallel to the line 3x − 4y + 5 = 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 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.
If the point (5, 2) bisects the intercept of a line between the axes, then its equation 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
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 equation of the line passing through the point (1, 2) and perpendicular to the line x + y + 1 = 0 is ______.
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).
