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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
If the angle between the lines represented by ax2 + 2hxy + by2 = 0 is equal to the angle between the lines 2x2 − 5xy + 3y2 = 0, then show that 100(h2 − ab) = (a + b)2
Concept: Angle between lines represented by ax2 + 2hxy + by2 = 0
Equation of line passing through the points (0, 0, 0) and (2, 1, –3) is ______.
Concept: General Second Degree Equation
Write the separate equations of lines represented by the equation 5x2 – 9y2 = 0
Concept: Combined Equation of a Pair Lines
Find the value of k. if 2x + y = 0 is one of the lines represented by 3x2 + kxy + 2y2 = 0
Concept: Homogeneous Equation of Degree Two
Find the vector equation of the lines passing through the point having position vector `(-hati - hatj + 2hatk)` and parallel to the line `vecr = (hati + 2hatj + 3hatk) + λ(3hati + 2hatj + hatk)`.
Concept: Equation of a Line in Space
Write the joint equation of co-ordinate axes.
Concept: Combined Equation of a Pair Lines
If ax2 + 2hxy + by2 = 0 represents a pair of lines and h2 = ab ≠ 0 then find the ratio of their slopes.
Concept: Angle between lines represented by ax2 + 2hxy + by2 = 0
If θ is the acute angle between the lines represented by ax2 + 2hxy + by2 = 0 then prove that tan θ = `|(2sqrt(h^2 - ab))/(a + b)|`
Concept: Angle between lines represented by ax2 + 2hxy + by2 = 0
Prove that the acute angle θ between the lines represented by the equation ax2 + 2hxy+ by2 = 0 is tanθ = `|(2sqrt(h^2 - ab))/(a + b)|` Hence find the condition that the lines are coincident.
Concept: Angle between lines represented by ax2 + 2hxy + by2 = 0
If the lines
`(x-1)/-3=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-5)/1=(z-6)/-5`
are at right angle then find the value of k
Concept: Shortest Distance Between Two Lines
Find the shortest distance between the lines
`bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)`
and
`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`
where λ and μ are parameters
Concept: Shortest Distance Between Two Lines
Show that the points (1, 1, 1) and (-3, 0, 1) are equidistant from the plane `bar r (3bari+4barj-12bark)+13=0`
Concept: Distance of a Point from a Plane
Show that the lines ` (x+1)/-3=(y-3)/2=(z+2)/1; ` are coplanar. Find the equation of the plane containing them.
Concept: Coplanarity of Two Lines
Find the equation of the planes parallel to the plane x + 2y+ 2z + 8 =0 which are at the distance of 2 units from the point (1,1, 2)
Concept: Distance of a Point from a Plane
Find the shortest distance between the lines `(x+1)/7=(y+1)/(-6)=(z+1)/1 and (x-3)/1=(y-5)/(-2)=(z-7)/1`
Concept: Shortest Distance Between Two Lines
Show that the points (1, –1, 3) and (3, 4, 3) are equidistant from the plane 5x + 2y – 7z + 8 = 0
Concept: Distance of a Point from a Plane
