Let y^2 4ax be a parabola and P be a point on it. Let the normal at P intersect the x-axis at Q. Draw a line at Q perpendicular to the above normal. Show that this line intersects the parabola y^2 =0.

Show that the coplanar points form a parallelogram.

Show that the point 3 lies on the ellipse with foci and and semi-major axis 3.

Show that the cone whose vertex is at the origin and which passes through the curve of intersection of the sphere x^2 y^2 z^2 and any plane which is at a distance from the origin has mutually perpendicular generators.

Find the equations of the spheres which pass through the circle
x^2 y^2 z^2 6x Z and touch the plane z =0.

Identify the conicoid 1 2x^2 9y^2 Does the xy-plane intersect with it Justify your answer.

Find the new equation of straight line 2x y 5 after rotating the axes through 45°.

Show that the conicoid
x^2 2y^2 2yz 2x 4y 8z 1 is central. Find the new equation of the conicoid if the origin is shifted to its centre.

5. State whether the following statements are true or false. Justify your answer with a short-explanation or a counter-example.

The plane making intercept 1 at the z-axis and parallel to the xy-plane intersects the cone x^2 y^2=z^2tan^28 in a circle.

There exists no line with as direction cosines.

The curve xy^2 yx^2 is symmetric about the origin.

The plane 3x 4y 2z 1 touches the conicoid 3x^2 2y^2 z^2 =1.

Non-degenerate conics are non-central.