Analytic Geometry Answers

What kind of conic section the following quadratic form represents and transform it to principal axes.

4x21 + 6x1x2 − 4x22 = 10.

Find the section of the conicoid (x² /2) − (y ²/ 3 )= 2z by the plane x− 2y+ z = 1. What conic does this section represent? Justify your answer.

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?

Prove that the path traced by the foot of the perpendicular from the focus of a parabola on any tangent to the parabola is the tangent at its vertex.

Find the equation of the right circular cone whose vertex is (1,−1,2), the axis is

(x−1 /2) =( y+1 /1) = (z−2 /−2)

and the semi-vertical angle is 45◦ .

Prove that the planes

7x+ 4y− 4z+ 30 = 0,36x−51y+12z+17 = 0,14x+8y−8z−12 = 0 and 12x−17y+4z−3 = 0 form the four faces of a cuboid.

please solve this ASAP.

Show that if ux+ vy+ wz = p is a tangent plane to the paraboloid ax² +by² = 2z, then

(u²/a) + (v²/b) + 2pw = 0.

Show that the angle between the two lines in which the plane x−y+2z = 0 intersects the cone

x² +y² −4z² +6yz = 0 is tan ⁻¹ (√ 6 /7)

Find the vertices, eccentricity, foci and asymptotes of the hyperbola (x^2/8) − (y² /4) = 1. Also trace it. Under what conditions on λ the line x+λy = 2 will be tangent to this hyperbola? Explain geometrically.

Find the equation of the line which passes through the point (1, √ 3) and makes an angle of 30◦ with the line x− √ 3y + √ 3 = 0