Question

Q1) Let S ≡ 4x² −9y²−36 = 0 and S' ≡ y² −4x = 0 be two conics. Under what

conditions on k, will the conic S+kS' = 0 represent:

i) an ellipse?

ii) a hyperbola?

Q2) Find the section of the conicoid x²/2-y²/3= 2z by the plane x−2y+z = 1. What

conic does this section represent? Justify your answer. 

Q3)If 1,1/2,0 are direction ratios of a line, then the line makes an angle of 90◦ with the

x-axis, an angle of 60◦ with the y-axis, and is parallel to the z-axis.

Q4)If a cone has three mutually perpendicular generators then its reciprocal cone has

three mutually perpendicular tangent planes.

Answer 1

1)

The conic $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$

is an ellipse if $B^2<4AC$ and a hyperbola if $B^2-4AC>0$

$S+kS'=4x^2+0xy+(k-9)y^2-4kx+0y-36=0$

is an ellipse if

02<4(4)(k-9)

0<16k-144

144<16k

k>9

an a hyperbola if

02-4(4)(k-9)>0

0-16k+144>0

-16k>-144

k<9

2)

$\frac{x^2}{2}-\frac{y^2}{3}=2z \\ x-2y+z=1\\ z=1-x+2y \\ \frac{x^2}{2}-\frac{y^2}{3}=2(1-x+2z)\\ (\sqrt{3}x+2\sqrt{3})^2-(\sqrt{2}y+3\sqrt{2})^2=42\\ \frac{(x+2)^2}{14}-\frac{(y+3)^2}{21}=1$

its a hyperbola

3)

If 1,1/2,0 are direction ratios of a line, then the line makes an angle of 90◦ with the

x-axis, an angle of 60◦ with the y-axis, and is parallel to the z-axis.

=True

4)

(l,m,n) are DC's of axes then

$l=m=n=\cos\theta=\left(\frac13\right)^\frac12$

$\tanθ=(2)^\frac12\implies θ=\arctan(2^\frac12)$