Advertisements
Advertisements
प्रश्न
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`
Advertisements
उत्तर
Given equations are x – y + 1 = 0 ......(i)
And 2x – 3y + 5 = 0 ......(ii)
Solving equation (i) and equation (ii) we get
2x – 2y + 2 = 0
2x – 3y + 5 = 0
(–) (+) (–)
y – 3 = 0
∴ y = 3
From equation (i) we have
x – 3 + 1 = 0
⇒ x = 2
So, (2, 3) is the point of intersection of equation (i) and equation (ii).
Let m be the slope of the required line
∴ Equation of the line is y – 3 = m(x – 2)
⇒ y – 3 = mx – 2m
⇒ mx – y + 3 – 2m = 0
Since, the perpendicular distance from (3, 2) to the line is `7/5`
Then `7/5 = |(m(3) - 2 + 3 - 2m)/sqrt(m^2 + 1)|`
Since, the perpendicular distance from (3, 2) to the line is `49/25 = ((3m - 2 + 3 - 2m)^2)/(m^2 + 1)`
⇒ `49/25 = (m + 1)^2/(m^2 + 1)`
⇒ 49m2 + 49 = 25m2 + 50m + 25
⇒ 49m2 – 25m2 – 50m + 49 – 25 = 0
⇒ 24m2 – 50m + 24 = 0
⇒ 12m2 – 25m + 12 = 0
⇒ 12m2 – 16m – 9m + 12 = 0
⇒ 4m(3m – 4) – 3(3m – 4) = 0
⇒ (3m – 4)(4m – 3) = 0
⇒ 3m – 4 = 0 and 4m – 3 = 0
∴ m = `4/3,3/4`
Equation of the line taking m = `4/3` is
y – 3 = `4/3(x - 2)`
⇒ 3y – 9 = 4x – 8
⇒ 4x – 3y + 1 = 0
Equation of the line taking m = `3/4` is
y – 3 = `3/4(x - 2)`
⇒ 4y – 12 = 3x – 6
⇒ 3x – 4y + 6 = 0
Hence, the required equations are 4x – 3y + 1 = 0 and 3x – 4y + 6 = 0
APPEARS IN
संबंधित प्रश्न
Find the equation of the line parallel to x-axis and having intercept − 2 on y-axis.
Draw the lines x = − 3, x = 2, y = − 2, y = 3 and write the coordinates of the vertices of the square so formed.
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 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 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 :
(0, 0) and (2, −2)
Find the equation of the straight lines passing through the following pair of point :
(a, b) and (a + b, a − b)
Find the equation of the straight lines passing through the following pair of point :
(at1, a/t1) and (at2, a/t2)
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 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').
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 the line which passes through the point (− 4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5 : 3 by this point.
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 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.
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 a line passing through (3, −2) and perpendicular to the line x − 3y + 5 = 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 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 length of the perpendicular from the origin to the straight line joining the two points whose coordinates are (a cos α, a sin α) and (a cos β, a sin β).
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 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).
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.
Find the equation of the straight line drawn through the point of intersection of the lines x + y = 4 and 2x − 3y = 1 and perpendicular to the line cutting off intercepts 5, 6 on the axes.
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 equation of the line passing through the point (1, −2) and cutting off equal intercepts from the axes.
Equation of the line passing through the point (a cos3θ, a sin3θ) and perpendicular to the line x sec θ + y cosec θ = a is x cos θ – y sin θ = a sin 2θ.
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).
