Advertisements
Advertisements
प्रश्न
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.
उत्तर
Given that: 2x + y = 5 .....(i)
x + 3y + 8 = 0 ......(ii)
3x + 4y = 7 ......(iii)
Equation of any line passing through the point of intersection of equation (i) and equation (ii) is
(2x + y – 5) + λ(x + 3y + 8) = 0 ......(iv) (λ = constant)
⇒ 2x + y – 5 + λx + 3λy + 8λ = 0
⇒ (2 + λ)x + (1 + 3λ)y – 5 + 8λ = 0
Slope of line m1 (say) = `(-(2 + lambda))/(1 + 3lambda)` .....`[because "m" = (-"a")/"b"]`
Now slope of line 3x + 4y = 7 is m2 (say) = `- 3/4`
If equation (iii) is parallel to equation (iv) then
m1 = m2
⇒ `(-(2 + lambda))/(1 + 3lambda) = - 3/4`
⇒ `(2 + lambda)/(1 + 3lambda) = 3/4`
⇒ 8 + 4λ = 3 + 9λ
⇒ 9λ – 4λ = 5
⇒ 5λ = 5
⇒ λ = 1
On putting the value of λ in equation (iv) we get
(2x + y – 5) + 1(x + 3y + 8) = 0
⇒ 2x + y – 5 + x + 3y + 8 = 0
⇒ 3x + 4y + 3 = 0
Hence, the required equation is 3x + 4y + 3 = 0.
APPEARS IN
संबंधित प्रश्न
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 (0, 0) with slope m.
Find the equation of the straight lines passing through the following pair of point :
(a, b) and (a + b, a − b)
By using the concept of equation of a line, prove that the three points (−2, −2), (8, 2) and (3, 0) are collinear.
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 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 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 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 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.
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.
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.
Three sides AB, BC and CA of a triangle ABC are 5x − 3y + 2 = 0, x − 3y − 2 = 0 and x + y − 6 = 0 respectively. Find the equation of the altitude through the vertex A.
Find the equation of a line passing through (3, −2) and perpendicular to the line x − 3y + 5 = 0.
Find the equation of the straight line through the point (α, β) and perpendicular to the line lx + my + n = 0.
Find the equation of the straight line perpendicular to 5x − 2y = 8 and which passes through the mid-point of the line segment joining (2, 3) and (4, 5).
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 of the straight lines passing through (2, −1) and making an angle of 45° with the line 6x + 5y − 8 = 0.
Find the equations of the two straight lines through (1, 2) forming two sides of a square of which 4x+ 7y = 12 is one diagonal.
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.
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 straight line which passes through the point (−4, 3) such that the portion of the line between the axes is divided internally by the point in the ratio 5 : 3 is
The equation of the line passing through (1, 5) and perpendicular to the line 3x − 5y + 7 = 0 is
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 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 ______.
