Advertisements
Advertisements
Question
A point equidistant from the lines 4x + 3y + 10 = 0, 5x – 12y + 26 = 0 and 7x + 24y – 50 = 0 is ______.
Options
(1, –1)
(1, 1)
(0, 0)
(0, 1)
Solution
A point equidistant from the lines 4x + 3y + 10 = 0, 5x – 12y + 26 = 0 and 7x + 24y – 50 = 0 is (0, 0).
Explanation:
Given equations are 4x + 3y + 10 = 0 .....(i)
5x – 12y + 26 = 0 ......(ii)
And 7x + 24y – 50 = 0 ......(iii)
Let (x1, y1) be any point equidistant from equation (i), equation (ii) and equation (iii).
Distance of (x1, y1) from equation (i)
= `|(4x_1 + 3y_1 + 10)/sqrt(16 + 9)|`
= `|(4x_1 + 3y_1 + 10)/5|`
Distance of (x1, y1) from equation (ii)
= `|(5x_1 - 12y_1 + 26)/sqrt(25 + 144)|`
= `|(5x_1 + 12y_1 + 26)/13|`
Distance of (x1, y1) from equation (iii)
= `|(7x_1 + 24y_1 - 50)/sqrt(49 + 576)|`
= `|(7x_1 + 24y_1 - 50)/25|`
If the point (x1, y1) is equidistant from the given lines, then
`|(4x_1 + 3y_1 + 10)/5| = |(5x_1 - 12y_1 + 26)/13|`
= `|(7x_1 + 2y_1 - 50)/25|`
We see that putting x1 = 0 and y1 = 0, the above relation is satisfied
i.e., `10/5 = 26/13 = 50/25` = 2
APPEARS IN
RELATED QUESTIONS
If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k
Find the points on the x-axis, whose distances from the `x/3 +y/4 = 1` are 4 units.
What are the points on the y-axis whose distance from the line `x/3 + y/4 = 1` is 4 units.
Find perpendicular distance from the origin to the line joining the points (cosΘ, sin Θ) and (cosΦ, sin Φ).
Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines x– 7y + 5 = 0 and 3x + y = 0.
Find the equation of the line whose perpendicular distance from the origin is 4 units and the angle which the normal makes with the positive direction of x-axis is 15°.
Find the distance of the point (3, 5) from the line 2x + 3y = 14 measured parallel to a line having slope 1/2.
The perpendicular distance of a line from the origin is 5 units and its slope is − 1. Find the equation of the line.
Find the equation of a line perpendicular to the line \[\sqrt{3}x - y + 5 = 0\] and at a distance of 3 units from the origin.
Find the distance of the point (4, 5) from the straight line 3x − 5y + 7 = 0.
Find the perpendicular distance of the line joining the points (cos θ, sin θ) and (cos ϕ, sin ϕ) from the origin.
Find the distance of the point of intersection of the lines 2x + 3y = 21 and 3x − 4y + 11 = 0 from the line 8x + 6y + 5 = 0.
Show that the product of perpendiculars on the line \[\frac{x}{a} \cos \theta + \frac{y}{b} \sin \theta = 1\] from the points \[( \pm \sqrt{a^2 - b^2}, 0) \text { is }b^2 .\]
Show that the path of a moving point such that its distances from two lines 3x − 2y = 5 and 3x + 2y = 5 are equal is a straight line.
If the length of the perpendicular from the point (1, 1) to the line ax − by + c = 0 be unity, show that \[\frac{1}{c} + \frac{1}{a} - \frac{1}{b} = \frac{c}{2ab}\] .
The equations of two sides of a square are 5x − 12y − 65 = 0 and 5x − 12y + 26 = 0. Find the area of the square.
Find the equations of the lines through the point of intersection of the lines x − y + 1 = 0 and 2x − 3y+ 5 = 0, whose distance from the point(3, 2) is 7/5.
Write the value of θ ϵ \[\left( 0, \frac{\pi}{2} \right)\] for which area of the triangle formed by points O (0, 0), A (a cos θ, b sin θ) and B (a cos θ, − b sin θ) is maximum.
If the lines x + ay + a = 0, bx + y + b = 0 and cx + cy + 1 = 0 are concurrent, then write the value of 2abc − ab − bc − ca.
The area of a triangle with vertices at (−4, −1), (1, 2) and (4, −3) is
The vertices of a triangle are (6, 0), (0, 6) and (6, 6). The distance between its circumcentre and centroid is
Find the distance between the lines 3x + 4y = 9 and 6x + 8y = 15.
The distance of the point P(1, – 3) from the line 2y – 3x = 4 is ______.
A point moves such that its distance from the point (4, 0) is half that of its distance from the line x = 16. The locus of the point is ______.
Find the points on the line x + y = 4 which lie at a unit distance from the line 4x + 3y = 10.
The ratio in which the line 3x + 4y + 2 = 0 divides the distance between the lines 3x + 4y + 5 = 0 and 3x + 4y – 5 = 0 is ______.
A straight line passes through the origin O meet the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at points P and Q respectively. Then, the point O divides the segment Q in the ratio:
