Advertisements
Advertisements
प्रश्न
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.
उत्तर
Let the given line x + y = 4 and required line ‘l’ intersect at B(a, b).
Slope of line ‘l’ is given by m = `(b - 2)/(a - 1)` = tan θ .....(i)
Given that AB = `sqrt(6)/3`
So, by distance formula for point A(1, 2) and B(a, b), we get
`sqrt((a - 1)^2 + (b - 2)^2) = sqrt(6)/3`
On squaring both the side
a2 + 1 – 2a + b2 + 4 – 4b = `6/9`
a2 + b2 – 2a – 4b + 5 = `2/3` .....(ii)
Point B(a, b) also satisfies the eqn. x + y = 4
∴ a + b = 4 .....(iii)
On solving (ii) and (iii)
We get a = `(3sqrt(3) + 1)/(2sqrt(3))`
b = `(5sqrt(3) - 1)/(2sqrt(3))`
Putting values of a and b in eqn. (i), we have
tan θ = `((5sqrt(3) - 1)/(2sqrt(3)))/((3sqrt(3) + 1)/(2sqrt(3))`
= `(5sqrt(3) - 1 - 4sqrt(3))/(3sqrt(3) + 1 - 2sqrt(3))`
= `(sqrt(3) - 1)/(sqrt(3) + 1)`
∴ tan θ = tan 15°
⇒ θ = 15°
APPEARS IN
संबंधित प्रश्न
Find the equation of the line parallel to x-axis and passing through (3, −5).
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 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 :
(at1, a/t1) and (at2, a/t2)
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 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 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 line, which passes through P (1, −7) and meets the axes at A and Brespectively so that 4 AP − 3 BP = 0.
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 \[\frac{AP}{BP} = \frac{2}{3}\] .
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.
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.
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 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 to the straight lines which pass through the origin and are inclined at an angle of 75° to the straight line \[x + y + \sqrt{3}\left( y - x \right) = a\].
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 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.
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.
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
If a + b + c = 0, then the family of lines 3ax + by + 2c = 0 pass through fixed point
The equation of the line passing through (1, 5) and perpendicular to the line 3x − 5y + 7 = 0 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
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.
