Advertisements
Advertisements
Question
Find the equation of the straight line which passes through the point of intersection of the lines 3x − y = 5 and x + 3y = 1 and makes equal and positive intercepts on the axes.
Advertisements
Solution
The equation of the straight line passing through the point of intersection of 3x − y = 5 and x + 3y = 1 is
3x − y − 5 + λ(x + 3y − 1) = 0
\[\Rightarrow\] (3 + λ)x + (−1 + 3λ)y − 5 − λ = 0 ... (1)
\[\Rightarrow y = - \left( \frac{3 + \lambda}{- 1 + \lambda} \right)x + \frac{5 + \lambda}{- 1 + \lambda}\]
The slope of the line that makes equal and positive intercepts on the axis is −1.
From equation (1), we have:
\[- \frac{3 + \lambda}{- 1 + 3\lambda} = - 1\]
\[ \Rightarrow \lambda = 2\]
Substituting the value of λ in (1), we get the equation of the required line.
\[\Rightarrow \left( 3 + 2 \right)x + \left( - 1 + 6 \right)y - 5 - 2 = 0\]
\[ \Rightarrow 5x + 5y - 7 = 0\]
APPEARS IN
RELATED QUESTIONS
Find the equation of the straight line which passes through the point (1,2) and makes such an angle with the positive direction of x-axis whose sine is \[\frac{3}{5}\].
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 equations to the altitudes of the triangle whose angular points are A (2, −2), B (1, 1) and C (−1, 0).
Find the equation of the straight lines passing through the following pair of point :
(0, −a) and (b, 0)
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).
By using the concept of equation of a line, prove that the three points (−2, −2), (8, 2) and (3, 0) are collinear.
In what ratio is the line joining the points (2, 3) and (4, −5) divided by the line passing through the points (6, 8) and (−3, −2).
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.
The owner of a milk store finds that he can sell 980 litres milk each week at Rs 14 per liter and 1220 liters of milk each week at Rs 16 per liter. Assuming a linear relationship between selling price and demand, how many liters could he sell weekly at Rs 17 per liter.
Find the equations to the straight lines which go through the origin and trisect the portion of the straight line 3 x + y = 12 which is intercepted between the axes of coordinates.
Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.
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.
The straight line through P (x1, y1) inclined at an angle θ with the x-axis meets the line ax + by + c = 0 in Q. Find the length of PQ.
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.
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.
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 drawn perpendicular to the line \[\frac{x}{4} + \frac{y}{6} = 1\] through the point where it meets the y-axis.
Find the equations to the straight lines which pass through the point (h, k) and are inclined at angle tan−1 m to the straight line y = mx + c.
Find the equations to the straight lines passing through the point (2, 3) and inclined at and angle of 45° to the line 3x + y − 5 = 0.
Find the equations to the sides of an isosceles right angled triangle the equation of whose hypotenues is 3x + 4y = 4 and the opposite vertex is the point (2, 2).
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.
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.
The equation of the base of an equilateral triangle is x + y = 2 and its vertex is (2, −1). Find the length and equations of its sides.
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.
Find the equation of lines passing through (1, 2) and making angle 30° with y-axis.
The equation of the line passing through the point (1, 2) and perpendicular to the line x + y + 1 = 0 is ______.
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 ______.
If a, b, c are in A.P., then the straight lines ax + by + c = 0 will always pass through ______.
