RGU Question Paper Mathematics Semester II 2022: Analytic Geometry

 

2022 (May/June )

BA/B.Sc (Il Semester) Examination (CBCS)

MATHEMATICS

Paper : MATH-CC-122 ( Analytic Geometry )

Full Marks : 80 

Pass Marks : 35%

Time : Three Hours

Note: .1 Questions are divided into three Sections A , B and C.

2. Read the instructions given in each Section carefully and answer the questions accordingly.

3. The figures in the margin indicate full marks for the questions.

Section-A

1. Answer any four parts: 5×4=20

(a) If by change of axes, without change of origin, the expression $a{x}^2+2hxy+b{y}^2$
becomes $A{x}^2+2Hxy+B{y}^2$, prove that $a+b=A+B$ and $ab-h^2=AB-H^{2.}$ 

(b) Find the equation of the line $y=\sqrt{3}x$ when the axes are rotated through an angle of $\pi /3$.

(c) Find the condition that the line lx +my = c is tangent to the circle $x^2+y^2=a^2$.

(d) Find the equation of a plane which passes through the point (2, 1, 4) and is perpendicular to the planes 9x-7y+6z+48=0 and x+y-z=0.

(e) Find the coordinates of the point, where the line

meets the plane x- 2y +3z +4 = 0.

(f) Find the equation of the cone whose vertex is (0, 0, 0) and which passes through the curve of intersection of the plane lx+my+nz =p and $a{x}^2+b{y}^2+c{z}^2=1$.

Section-B 

2. Answer any three parts: 10×3=30

(a) (i) Find the equation of a cylinder whose generators are parallel to the line 

and whose guiding curve is 

5

(ii) Find the equation of the cone whose vertex is the point $(\alpha ,\beta ,\gamma )$

 and the base is the parabola $z=0, y^2=4ax$. 5

(b) (i) Find the equation of the sphere passing through (0, 0, 0), (a, 0, 0), (0, b, 0) and (0, 0, c). 4

(i) Find the shortest distance between the lines

Also, find the equation of the line of shortest distance.  6

(c) (i) Find the equation of the tangent to the conic

at the point (1, - 2).  4

(ii) Find the polar equation of a conic having eccentricity e, latus rectum 2l and focus being taken as pole.  6

(d) (i) Obtain the condition that the general second-degree equation 

represents a pair of straight lines. 5

(ii) Find the equation of the pair of lines through the origin and perpendicular to the pair of lines $a{x}^2+2hxy+b{y}^2=0$. 5

(e) (i) Find the equation of common tangent to the circle $x^2+y^2=4ax$ and the parabola $y^2=4ax$. 5

(ii) Write the equation of the line 3x- 4y+2z+5=0=2x+3y-52-8 in symmetrical form. 5

Section-C

3. Answer any two parts: 15×2=30

(a) (i) Prove that the lines

are coplanar. Find also the equation of the plane.  6 

(ii) Prove that the line x= pz +q, y=Pz+Q intersects the conic z=0, $a{x}^2+b{y}^2=1$ if

$a{q}^2+b{Q}^2=1$. 4

(iii) A plane passes through a fixed point (p, q, r) and cuts the coordinate axes. Show that the locus of the centre of the sphere is

5

(b) (i) Prove that the line

will touch the conic

if $(A-e{)}^2+B^2=1$, where e is the eccentricity. 6

(ii) Find the equation of the tangent and normal at (1,-1) to the conic 

4.

(iii) Reduce the equation

to standard form. 5

(c) (i) Find the value of K so that the equation

may represent a pair of straight lines.

(ii) Find the equation of the polar of the point (2, 3) with respect to the conic

(iii) Prove that the straight line

will be normal to the ellipse

if $5c=a^2{e}^2$ where e is the eccentricity of the ellipse.

(d) (i) Define a right circular cone. Find the equation of a right circular cone whose vertex is origin, axis is the z-axis and the semi-vertical angle is $\mathrm{\alpha }$.

(i) Prove that the equation of the right circular cylinder whose axis is

and radius r is given by

 5.

(ii) Prove that

represents two parallel straight lines if

6.