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If the normal at one extremity of latus rectum of an ellipse passes through one extremity of the minor axis, prove that the eccentricity e is given by e⁴+e²-1=0

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?

How to 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.

To 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 45º

x-1/2 = y+1/1 = z-2/-2

and the semi-vertical angle is 45º

The sketch shows the hyperbola defined by y the straight line defined by y ; a circle with centre at P, touching the r-axis and y-axis at R and S, respectively; and the straight line through the points T, S and R. The line joining T and Q is parallel to the y-axis. 2.1 Determine the coordinates of P and Q (5) 2.2 Determine the coordinates of S and R (2) 2.3 Find the radius of the circle, and write down the equation of the circle (3) 2.4 Determine the equation of the line through T, S and R. (5) 2.5 Calculate the length of the line TQ (5)

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.

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.

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 x²/8-y²/4=1

Also trace it. Under what conditions on λ the line x+λy=2 will be tangent to this hyperbola? Explain geometrically.

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 30º with the line x-√(3)y+√3=0