Analytic Geometry Answers

If the equation of a parabola with the focus at (3,−4) and the directrix x+y = 2 is
x
2 +y
2 −2xy−8x+20y+c = 0, then what is the value of c

If P is a point on an ellipse with a focus F, then PF is always greater than PM,
where M is the foot of the perpendicular drawn from P to a directrix of the ellipse.

The circle (x−1)
2 +y
2 = 1,z = 0 lies inside the sphere centred at the origin, and
having radius 2√
2.

Give the equation of a conic which is symmetric to the line x+2 = 0.

Find the equation of the hyperbola with vertices (1,−4) and (1,4), and foci at
(1,−6) and (1,6)

Derive the equation (23) at page 42 of Unit 2, which represents the polar equation
of a conic when the directrix L corresponding to a focus F is taken to the right of F.

The normals at any point P of the ellipsoid x
2
9 +
y
2
4 +z
2 = 1 meet the coordinate
planes in Q1,Q2,Q3, respectively. Show that PQ1 : PQ2 : PQ3 :: 9 : 4 : 1.

Find the equation of the normal to the solid 2x
2 −y
2 +8z
2 = 11 at a point where it
intersects the line x−3 = z =
y+1
2
.

Find the equation of the normal to the solid 2x
2 −y
2 +8z
2 = 11 at a point where it
intersects the line x−3 = z =
y+1
2
.

Check whether the points (1,−1,−2),(1,−4,2),(3,0,2),(4,−3−2) are coplanar
or not. If they are coplanar, write the equation of the plane they pass through.
Otherwise, change the coordinates of one of the points so that they become
coplanar. In this case, find the plane passing through them.