Advertisements
Advertisements
प्रश्न
If the length of the perpendicular from the point (1, 1) to the line ax − by + c = 0 be unity, show that \[\frac{1}{c} + \frac{1}{a} - \frac{1}{b} = \frac{c}{2ab}\] .
Advertisements
उत्तर
The distance of the point (1, 1) from the straight line ax − by + c = 0 is 1
\[\therefore 1 = \left| \frac{a - b + c}{\sqrt{a^2 + b^2}} \right|\]
\[ \Rightarrow a^2 + b^2 + c^2 - 2ab + 2ac - 2bc = a^2 + b^2 \]
\[ \Rightarrow ab + bc - ac = \frac{c^2}{2}\]
Dividing both the sides by abc, we get:
APPEARS IN
संबंधित प्रश्न
Find the distance between parallel lines:
15x + 8y – 34 = 0 and 15x + 8y + 31 = 0
Find the distance between parallel lines l (x + y) + p = 0 and l (x + y) – r = 0
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 co-ordinates of the point, which divides the line segment joining the points A(2, − 6, 8) and B(− 1, 3, − 4) externally in the ratio 1 : 3.
Prove that the line y − x + 2 = 0 divides the join of points (3, −1) and (8, 9) in the ratio 2 : 3.
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 equation of the straight line at a distance of 3 units from the origin such that the perpendicular from the origin to the line makes an angle tan−1 \[\left( \frac{5}{12} \right)\] with the positive direction of x-axi .
Find the distance of the point (3, 5) from the line 2x + 3y = 14 measured parallel to a line having slope 1/2.
Find the distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to a line having slope 3/4.
Find the distance of the point (3, 5) from the line 2x + 3y = 14 measured parallel to the line x − 2y = 1.
Find the distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to the line 3x − 4y+ 8 = 0.
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.
What are the points on X-axis whose perpendicular distance from the straight line \[\frac{x}{a} + \frac{y}{b} = 1\] is a ?
Find the perpendicular distance from the origin of the perpendicular from the point (1, 2) upon the straight line \[x - \sqrt{3}y + 4 = 0 .\]
What are the points on y-axis whose distance from the line \[\frac{x}{3} + \frac{y}{4} = 1\] is 4 units?
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.
Determine the distance between the pair of parallel lines:
4x − 3y − 9 = 0 and 4x − 3y − 24 = 0
Determine the distance between the pair of parallel lines:
y = mx + c and y = mx + d
Determine the distance between the pair of parallel lines:
4x + 3y − 11 = 0 and 8x + 6y = 15
The equations of two sides of a square are 5x − 12y − 65 = 0 and 5x − 12y + 26 = 0. Find the area of the square.
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.
Write the distance between the lines 4x + 3y − 11 = 0 and 8x + 6y − 15 = 0.
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.
Write the locus of a point the sum of whose distances from the coordinates axes is unity.
L is a variable line such that the algebraic sum of the distances of the points (1, 1), (2, 0) and (0, 2) from the line is equal to zero. The line L will always pass through
The area of a triangle with vertices at (−4, −1), (1, 2) and (4, −3) is
The line segment joining the points (1, 2) and (−2, 1) is divided by the line 3x + 4y = 7 in the ratio ______.
Distance between the lines 5x + 3y − 7 = 0 and 15x + 9y + 14 = 0 is
The vertices of a triangle are (6, 0), (0, 6) and (6, 6). The distance between its circumcentre and centroid is
A plane passes through (1, - 2, 1) and is perpendicular to two planes 2x - 2y + z = 0 and x - y + 2z = 4. The distance of the plane from the point (1, 2, 2) is ______.
If the tangent to the curve y = 3x2 - 2x + 1 at a point Pis parallel toy = 4x + 3, the co-ordinates of P are
If the sum of the distances of a moving point in a plane from the axes is 1, then find the locus of the point.
The distance of the point of intersection of the lines 2x – 3y + 5 = 0 and 3x + 4y = 0 from the line 5x – 2y = 0 is ______.
The distance between the lines y = mx + c1 and y = mx + c2 is ______.
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 ______.
