Advertisements
Advertisements
Question
Prove that the line y − x + 2 = 0 divides the join of points (3, −1) and (8, 9) in the ratio 2 : 3.
Advertisements
Solution
Let y − x + 2 = 0 divide the line joining the points (3, −1) and (8, 9) at point P in the ratio k : 1
\[\therefore P \equiv \left( \frac{3 + 8k}{k + 1}, \frac{- 1 + 9k}{k + 1} \right)\]
P lies on the line y − x + 2 = 0
\[\therefore \frac{- 1 + 9k}{k + 1} - \frac{3 + 8k}{k + 1} + 2 = 0\]
\[ \Rightarrow - 1 + 9k - 3 - 8k + 2k + 2 = 0\]
\[ \Rightarrow 3k = 2\]
\[ \Rightarrow k = \frac{2}{3}\]
Hence, the line y − x + 2 = 0 divides the line joining the points (3, −1) and (8, 9) in the ratio 2 : 3
APPEARS IN
RELATED QUESTIONS
Find the points on the x-axis, whose distances from the `x/3 +y/4 = 1` are 4 units.
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 direction in which a straight line must be drawn through the point (–1, 2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.
If sum of the perpendicular distances of a variable point P (x, y) from the lines x + y – 5 = 0 and 3x – 2y+ 7 = 0 is always 10. Show that P must move on a line.
A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.
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.
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°.
A line passes through a point A (1, 2) and makes an angle of 60° with the x-axis and intersects the line x + y = 6 at the point P. Find AP.
Find the distance of the point (2, 3) from the line 2x − 3y + 9 = 0 measured along a line making an angle of 45° with the x-axis.
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 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.
Show that the perpendiculars let fall from any point on the straight line 2x + 11y − 5 = 0 upon the two straight lines 24x + 7y = 20 and 4x − 3y − 2 = 0 are equal to each other.
What are the points on y-axis whose distance from the line \[\frac{x}{3} + \frac{y}{4} = 1\] is 4 units?
If sum of perpendicular distances of a variable point P (x, y) from the lines x + y − 5 = 0 and 3x − 2y + 7 = 0 is always 10. Show that P must move on a line.
Determine the distance between the pair of parallel lines:
8x + 15y − 34 = 0 and 8x + 15y + 31 = 0
Find the ratio in which the line 3x + 4y + 2 = 0 divides the distance between the line 3x + 4y + 5 = 0 and 3x + 4y − 5 = 0
If the centroid of a triangle formed by the points (0, 0), (cos θ, sin θ) and (sin θ, − cos θ) lies on the line y = 2x, then write the value of tan θ.
The area of a triangle with vertices at (−4, −1), (1, 2) and (4, −3) is
The value of λ for which the lines 3x + 4y = 5, 5x + 4y = 4 and λx + 4y = 6 meet at a point is
The vertices of a triangle are (6, 0), (0, 6) and (6, 6). The distance between its circumcentre and centroid is
The ratio in which the line 3x + 4y + 2 = 0 divides the distance between the line 3x + 4y + 5 = 0 and 3x + 4y − 5 = 0 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 ______.
The shortest distance between the lines
`bar"r" = (hat"i" + 2hat"j" + hat"k") + lambda (hat"i" - hat"j" + hat"k")` and
`bar"r" = (2hat"i" - hat"j" - hat"k") + mu(2hat"i" + hat"j" + 2hat"k")` is
Find the distance between the lines 3x + 4y = 9 and 6x + 8y = 15.
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:
Find the length of the perpendicular drawn from the point P(3, 2, 1) to the line `overliner = (7hati + 7hatj + 6hatk) + λ(-2hati + 2hatj + 3hatk)`
