MAT2131 - Coordinate Geometry and Vector Analysis¶

Coordinate and Polar System¶

Q1(a) [4]: Find the equation of the plane through the points \((1,-2,2)\) and \((-3,1,2)\) and perpendicular to plane \(2x+y-z+6=0\).
Q1(b) [5]: Remove first degree terms from the equation \(3x^2+4y^2-12x+4y+13=0\).
Q1(c) [5]: Find the condition that the general equation \(ax^2+2hxy+by^2+2gx+2fy+c=0\) may represents a pair of straight lines.

Q1(a) [3]: Find the relation between Cartesian coordinates and Polar coordinates.
Q1(b) [3]: Show that the area of the triangle formed by the straight lines \(y-2x=0\), \(y-3x=0\) and \(y=5x+4\) is \(\frac{4}{3}\).
Q1(c) [4]: Prove that a homogeneous quadratic equation \(ax^2+2hxy+by^2=0\) always represents a pair of straight lines which passing through the origin.
Q1(d) [4]: Show that the lines joining the origin to the points of intersection of \(7x^2+8xy-7y^2+6x-12y=0\) and \(2x+y-1=0\) are at right angles.
Q2(a) [5]: Prove that the straight lines represented by the equation \(ax^2+2hxy+by^2+2gx+2fy+c=0\) will be equidistance from the origin, if \(f^4-g^4=c(bf^2-ag^2)\).
Q2(b) [3]: Prove that the equation \(x^2+6xy+9y^2+4x+12y-5=0\) represent a pair of parallel lines.
Q3(a) [3]: Find the value of \(\lambda\) so that the equation \(\lambda x^2+4xy+y^2-4x-2y-3=0\) may represent a pair of straight lines.
Q3(b) [3]: If the pair of straight lines \(x^2-2mxy-y^2=0\) and \(x^2-2nxy-y^2=0\) be such that each pair bisects the angle between the other pair, prove that \(mn+1=0\).

Q1(a): Find the condition that the general equation of the second degree \(ax^2+2hxy+by^2+2gx+2fy+c=0\) may represents a pair of straight lines.
Q1(b): By transforming to parallel axes through a properly chosen point \((h,k)\), prove that the equation \(12x^2-10xy+2y^2+11x-5y+2=0\) can be reduced to one containing only the terms of the second degree.
Q1(c): Show that the equation \(bx^2-2hxy+ay^2=0\) represents a pair of straight lines which are at right angle to the pair given by equation \(ax^2+2hxy+by^2=0\).
Q2(a): Find the value of k so that the equation \(6x^2+2kxy+12y^2+22x+31y+20=0\) may represent a pair of straight lines.
Q3(a): Test the nature of the conic given by the equation \(3x^2-8xy-3y^2+10x-13y+8=0\).
Q3(b): Find the equation of the sphere through the points \((0,0,0)\), \((1,-1,0)\), \((2,0,-2)\) and \((0,1,2)\).

Q1(a) [3]: Find the relation between Cartesian coordinate and polar coordinate.
Q1(b) [5]: Origin shifted at \((4,-6)\). Find transformation of equation \(3x^2+5y^2+10xy+10=0\).
Q1(c) [6]: If the direction of axes is turned through an angle \(45^\circ\) and the origin remains unchanged then find the transformation equation of \(x^2-2xy+y^2+2x-4y+3=0\).

Q1(a) [4]: Find the relation between Cartesian coordinates and Polar coordinates.
Q1(b) [4]: If the Axes are turned through \(45^\circ\), find the transformed form of the equation \(3x^2+3y^2+2xy=2\).
Q1(c) [6]: Show that the necessary condition of bisectors of the angles between the lines represented by \(ax^2+2hxy+by^2=0\) is \(\frac{x^2-y^2}{a-b}=\frac{xy}{h}\).
Q2(a) [3]: If the pair of straight lines \(x^2-2axy-y^2=0\) and \(x^2-2bxy-y^2=0\) be such that each pair bisects the angle between the other pair, prove that \(ab=-1\).
Q2(b) [5]: Prove that the straight lines represented by the equation \(ax^2+2hxy+by^2+2gx+2fy+c=0\) will be equidistance from the origin, if \(f^4-g^4=c(bf^2-ag^2)\).
Q3(a) [6]: Define circle. Show that the four points \((1,0)\), \((2,-7)\), \((8,1)\) and \((9,-6)\) are concyclic.
Q3(b) [3]: If \(ax^2+2hxy+by^2+2gx+2fy+c=0\) represents the non-degenerate conic, then write down the natures of conic.
Q3(c) [5]: For what value of \(\lambda\) the equation \(2xy+4x-6y+\lambda=0\) represent two intersecting lines? If \(\lambda=17\), then what type of conic represents by the given equation?

Q1(a) [6]: Find the relation between Cartesian coordinates and polar coordinates. Transform to polar coordinates the equation \(x^3=y^2(2a-x)\).
Q1(b) [5]: Determine the equation of the parabola \(x^2-2xy+y^2+2x-4y+3=0\) after rotating of axes through \(45^\circ\).
Q1(c) [3]: Prove that a homogeneous quadratic equation \(ax^2+2hxy+by^2=0\) always represents a pair of straight lines which passing through the origin.
Q2(a) [3]: Show that the necessary condition of bisector of the angles between the lines represented by \(ax^2+2hxy+by^2=0\) is \(\frac{x^2-y^2}{a-b}=\frac{xy}{h}\).
Q2(b) [3+5]: Prove that the equation \(x^2+6xy+9y^2+4x+12y-5=0\) represent a pair of parallel lines.
Q2(c) [5]: Test the nature and find the center of the conic represented by \(2x^2-3xy+y^2-5x-4y+6=0\).
Q3(a): Find the area of the triangle formed by the lines \(ax^2+2hxy+by^2=0\) and \(lx+my+n=0\).
Q3(b): Find the equation of the sphere with its center at \((-4,2,3)\) and tangent to the plane \(2x-y+2z+7=0\).
Q3(c): Find the distance of the points \((2,0,1)\) and \((3,-3,2)\) from the plane \(x-2y+z=6\) and find whether the two points lie on the same side or opposite sides of the plane.