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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
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
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
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.
Concept: Distance of a Point from a Line
Find the distance of the point (1, 2, –1) from the plane x - 2y + 4z - 10 = 0 .
Concept: Distance of a Point from a Plane
A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.
Concept: Vector and Cartesian Equations of a Line
Show that the line `(x - 2)/(1) = (y - 4)/(2) = (z + 4)/(-2)` passes through the origin.
Concept: Vector and Cartesian Equations of a Line
Find the co-ordinates of the foot of the perpendicular drawn from the point `2hati - hatj + 5hatk` to the line `barr = (11hati - 2hatj - 8hatk) + λ(10hati - 4hatj - 11hatk).` Also find the length of the perpendicular.
Concept: Equation of a Plane
Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 6y – 3z = 63.
Concept: Equation of a Plane
Find the vector equation of the plane passing through the point having position vector `hati + hatj + hatk` and perpendicular to the vector `4hati + 5hatj + 6hatk`.
Concept: Equation of a Plane
Find the vector and Cartesian equations of the line passing through the point (–1, –1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z − 2.
Concept: Vector and Cartesian Equations of a Line
Choose correct alternatives:
The vector equation of line 2x – 1 = 3y + 2 = z – 2 is ______.
Concept: Vector and Cartesian Equations of a Line
The direction ratios of the line which is perpendicular to the two lines `(x - 7)/(2) = (y + 17)/(-3) = (z - 6)/(1) and (x + 5)/(1) = (y + 3)/(2) = (z - 4)/(-2)` are ______.
Concept: Vector and Cartesian Equations of a Line
Choose correct alternatives :
The shortest distance between the lines `vecr = (hati + 2hatj + hatk) + lambda(hati - hatj + hatk) and vecr = (2hati - hatj - hatk) + μ(2hati + hatj + 2hatk)` is ______.
Concept: Distance Between Skew Lines and Parallel Lines
The direction ratios of the line 3x + 1 = 6y – 2 = 1 – z are ______.
Concept: Distance Between Skew Lines and Parallel Lines
The foot of the perpendicular drawn from the origin to a plane is M(1, 2, 0). Find the vector equation of the plane.
Concept: Vector and Cartesian Equations of a Line
Find the cartesian equation of the plane passing through A(1, 2, 3) and the direction ratios of whose normal are 3, 2, 5.
Concept: Vector and Cartesian Equations of a Line
Find the direction ratios of the line perpendicular to the lines
`(x - 7)/2 = (y + 7)/(-3) = (z - 6)/1` and `(x + 5)/1 = (y + 3)/2 = (z - 6)/(-2)`
Concept: Vector and Cartesian Equations of a Line
