Course Code: MTE-05

Assignment Code: MTE-05/TMA/2023

Maximum Marks: 100

1. Check whether the following statements are true or false. Justify your answer with a short explanation or a counter example.

\((2 \times 10=20)\)

(i) The numbers \(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}, \frac{3}{4}\) are the direction cosines of a line.

(ii) The points \((1,2),(7,6)\) and \((4,4)\) are collinear.

(iii) The conic \(12 x^{2}+12 x y+3 y^{2}+2 x+y=0\) is degenerate.

(iv) Intersection of the ellipsoid \(\frac{x^{2}}{4}+\frac{y^{2}}{25}+\frac{z^{2}}{4}=1\) and the plane \(y=5\) is a circle.

(v) The conicoid \(3 x^{2}+y^{2}+2 x y+x-y-z+1=0\) is non-central.

(vi) The line \(y=x\) is a tangent to the parabola \(y^{2}=c x, c>0\).

(vii) The equation \(2 x^{2}+y+z+1=0\) represents a paraboloid.

(viii) Projection of a line segment on a line perpendicular to it is the length of the line segment.

(ix) The lines \(x=-y, z=2\) and \(x=y, z=0\) intersect each other.

(x) Every planar section of a cylinder is a circle.

2. (a) Trace the conic \(x^{2}-2 x y+y^{2}-3 x+2 y+3=0\).

(b) Prove that the conic passing through the points of intersection of two rectangular hyperbolas is also a rectangular hyperbola.

(c) Show that the line \(x=y\) touches the conic \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+\) \(c=0\), if \(f+g=0\).

3. (a) Let \(P\) be the midpoint of the line segment joining the points \(A(a+b, b)\) and \(B(a-b, a+b)\). Find the slope of the line passing through \(P\) and \(Q\left(b,-\frac{a}{2}\right)\). Under what conditions on \(a\) and \(b\), this line is parallel to the \(y\)-axis?

(b) (i) Show that \(\left|\begin{array}{ccc}x & y & 1 \\ 2 & 3 & 1 \\ -4 & 7 & 1\end{array}\right|=0\) represents the equation of a line passing through \((2,3)\) and \((-4,7)\)

(ii) Prove that the equation of a line through \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) can be expressed in the form \(\left|\begin{array}{lll}x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1\end{array}\right|=0\)

(c) Find the eccentricity, foci, centre and directrices of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\). Also give a rough sketch of it.

(d) Prove that the length of the chord of a parabola which passes through the focus and which is inclined at \(30^{\circ}\) to the axis of the parabola is four times the length of the latus rectum.

4. (a) Find the equations of the line through \((1,3,4)\) and parallel to the line joining the points \((-4,5,3)\) and \((8,9,7)\).

(b) Find the equation of the plane which passes through the line of intersection of the planes \(3 x+4 y-5 z=9\) and \(2 x+6 y+6 z=7\) and which is perpendicular to the plane \(3 x+2 y-5 z+6=0\). (c) Find the distance of the origin from the plane which passes through \((2,1,8),(1,0,2)\) and \((-3,4,6)\).

5. (a) Show that the plane \(2 x+y+2 z=0\) is a tangent plane to the sphere \(x^{2}+y^{2}+\) \(z^{2}-2 x+2 y-2 z+2=0\).

(b) Find the equation of the sphere touching the plane \(8 x+5 y+3 z+1=0\) at \((3,-1,-1)\) and cutting the sphere \(x^{2}+y^{2}+z^{2}-2 x+y-z-6=0\) orthogonally.

(c) Find the angle between the lines of intersection of the cone \(4 x^{2}+y^{2}+4 z^{2}+\) \(4 y z+2 z x=0\) and the plane \(x+2 y+3 z=0\).

(d) Find the equation of the cylinder with base

\[
x^{2}+y^{2}+z^{2}-3 x-6 z+9=0, x-2 y+2 z-6=0
\]

6. (a) Show that the perpendiculars drawn from the origin to tangent planes to the cone \(x^{2}-y^{2}+5 z^{2}+4 x y=0\) lie on the cone \(x^{2}-y^{2}+z^{2}+4 x y=0\).

(b) Transform the equation \(x^{2}+2 y^{2}-6 z^{2}-2 x-8 y+3=0\) by shifting the origin to \((1,2,0)\) without changing the directions of the coordinate axes. What object does this new equation represent? Give a rough sketch of it.

(c) Show that the conicoid \(2 x^{2}+2 y^{2}+x y-y z+z x+2 x-y+5 z+1=0\) is central. Hence find its centre.

7. (a) Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.

(i) \(x^{2}+y^{2}+z^{2}+4 x+3 y-z=0\)

(ii) \(2 x^{2}-y^{2}-z^{2}+x y+y z-z x=1\)

(iii) \(x^{2}+y^{2}-z^{2}-2 x y-3 y z-6 z x+x-2 y+5 z+4=0\)

(b) Find the transformation of the equation \(12 x^{2}-2 y^{2}+z^{2}=2 x y\) if the origin is kept fixed and the axes are rotated in such a way that the direction ratios of the new axes are \(1,-3,0 ; 3,1,0 ; 0,0,1\).

(c) Find the projection of the line segment joining the points \((1,-1,6)\) and \((4,3,2)\) on the line \(\frac{x-4}{3}=-y=\frac{z}{5}\).

8. (a) Identify and trace the conicoid \(y^{2}+3 z^{2}=x\). Describe its sections by the planes \(y=0\) and \(z=0\).

(b) Find the equation of tangent plane to the conicoid \(x^{2}+3 y^{2}=4 z\) at \((2,-4,13)\). Represent the tangent plane geometrically.