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Let `A(bara)` and `B(barb)` be any two points in the space and `R(barr)` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar r=(mbarb+nbara)/(m+n)`. Hence find the position vector of R which divides the line segment joining the points A(1, –2, 1) and B(1, 4, –2) internally in the ratio 2 : 1.
Concept: Equation of a Line in Space
Find the vector equation of the lines which passes through the point with position vector `4hati - hatj +2hatk` and is in the direction of `-2hati + hatj + hatk`
Concept: Equation of a Line in Space
The equation of a line is 2x -2 = 3y +1 = 6z -2 find the direction ratios and also find the vector equation of the line.
Concept: Equation of a Line in Space
Find the combined equation of the following pair of lines:
2x + y = 0 and 3x − y = 0
Concept: Combined Equation of a Pair Lines
Find the combined equation of the following pair of lines passing through point (2, 3) and parallel to the coordinate axes.
Concept: Combined Equation of a Pair Lines
Find the combined equation of the following pair of line passing through (−1, 2), one is parallel to x + 3y − 1 = 0 and other is perpendicular to 2x − 3y − 1 = 0
Concept: Combined Equation of a Pair Lines
Find the separate equation of the line represented by the following equation:
3y2 + 7xy = 0
Concept: Combined Equation of a Pair Lines
Find k, the slope of one of the lines given by kx2 + 4xy – y2 = 0 exceeds the slope of the other by 8.
Concept: Homogeneous Equation of Degree Two
If one of the lines given by ax2 + 2hxy + by2 = 0 bisects an angle between the coordinate axes, then show that (a + b)2 = 4h2.
Concept: Homogeneous Equation of Degree Two
Find the coordinates of the points of intersection of the lines represented by x2 − y2 − 2x + 1 = 0
Concept: General Second Degree Equation in x and y
If the lines represented by kx2 − 3xy + 6y2 = 0 are perpendicular to each other, then
Concept: Equation of a Line in Space
The area of triangle formed by the lines x2 + 4xy + y2 = 0 and x - y - 4 = 0 is ______.
Concept: Combined Equation of a Pair Lines
Find the joint equation of the line passing through the origin having slopes 2 and 3.
Concept: Combined Equation of a Pair Lines
Show that the difference between the slopes of the lines given by (tan2θ + cos2θ)x2 − 2xy tan θ + (sin2θ)y2 = 0 is two.
Concept: Angle between lines represented by ax2 + 2hxy + by2 = 0
The separate equations of the lines represented by `3x^2 - 2sqrt(3)xy - 3y^2` = 0 are ______
Concept: Equation of a Line in Space
The combined equation of the lines through origin and perpendicular to the pair of lines 3x2 + 4xy − 5y2 = 0 is ______
Concept: Combined Equation of a Pair Lines
Find the value of h, if the measure of the angle between the lines 3x2 + 2hxy + 2y2 = 0 is 45°.
Concept: Angle between lines represented by ax2 + 2hxy + by2 = 0
Show that the combined equation of pair of lines passing through the origin is a homogeneous equation of degree 2 in x and y. Hence find the combined equation of the lines 2x + 3y = 0 and x − 2y = 0
Concept: Combined Equation of a Pair Lines
If θ is the acute angle between the lines given by ax2 + 2hxy + by2 = 0 then prove that tan θ = `|(2sqrt("h"^2) - "ab")/("a" + "b")|`. Hence find acute angle between the lines 2x2 + 7xy + 3y2 = 0
Concept: Angle between lines represented by ax2 + 2hxy + by2 = 0
Find the joint equation of pair of lines through the origin which is perpendicular to the lines represented by 5x2 + 2xy - 3y2 = 0
Concept: Equation of a Line in Space
