Analytic Geometry Answers

Find the equation of the line through the origin with slope -6 Answer: 6x+y=0

Reduce the following equations to standard form, and then identify which conicoids they represent. Further, give a rough sketch of each.
i) x^2 +y^2 +2x−y−z+3 = 0
ii) 3y^2 +3z^2 +4x+3y+z = 9

Show that x = y = z+1 is a secant line of the sphere x^2 +y^2 +z^2 −x−y+z−1 = 0.
Also find the intercept made by the sphere on the line.

A right circular cylinder passes through the point (1,−1,4) and has the axis along
the line x−1/2 =y−3/5 =z+1/3. Is this information sufficient to determine the equation of
the cylinder? If it is, determine the equation of the cylinder. Otherwise, state another condition so that the equation can be determined uniquely, and also find the equation.

a) 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?
b) Find the equation of the plane passing through the line of intersection of the planes
2x+3y+z = 4 and x+y+z = 2, and which is perpendicular to the plane 2x+3y−z = 3.
c) What is the new equation of the conic x^2 +y^2 +4x−2y+3 = 0, when
i) the origin is shifted at (2,−1), followed by a rotation of axes through 45°?
ii) the axes are rotated through 45°, followed by the shifting of the origin at(2,−1)?
Are the equations in i) and ii) above the same?Why?
d) Does there pass a plane through the lines x+4/3 =y/2 =z−1/3
and x/2 =y−1/1 =z+1/1?Justify.
e) Find the equation of the right circular cone whose vertex is (1,0,1), the axis is
x−1 = y−2 = z−3, and the semi-vertical angle is 30°. Also, find the section of the cone by the coordinate planes.

Which of the following statements are true, and which are false? Justify your answer
with a short proof or a counter-example.
i) The set of all the points (x, y,z) satisfying the equation x−z = z−y represents a
line.
ii) Any two conics can intersect at at most two points.
iii) The circle (x−1)^2+y^2 = 1,z = 0 lies inside the sphere centred at the origin, and
having radius 2√2.
iv) If a curve C is symmetric about both the coordinate axes, then C is symmetric about
the origin.
v) Every planar section of an ellipsoid is an ellipse.
vi) If α,β, γ are direction ratios of a line, then so are α^2,β^2, γ^2.
vii) The projection of the line segment joining (1,2,−1) and (4,2,−1) on the line
x = y = z is 0.
viii) The polar equation r = θ represents a conic.
ix) 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.
x) x^2 +y^2 +z^2 = xyz is the equation of a cone.

Which of the following statements are true, and which are false? Justify your answer
with a short proof or a counter-example

i) The set of all the points (x, y,z) satisfying the equation x−z = z−y represents a
line.
ii) Any two conics can intersect at at most two points.
iii) The circle (x−1)²+y²= 1,z = 0 lies inside the sphere centred at the origin, and
having radius 2√2
iv) If a curve C is symmetric about both the coordinate axes, then C is symmetric about
the origin.
v) Every planar section of an ellipsoid is an ellipse.
vi) If α,β, γ are direction ratios of a line, then so are α²,β², γ²
.
vii) The projection of the line segment joining (1,2,−1) and (4,2,−1) on the line
x = y = z is 0.
viii) The polar equation r = θ represents a conic.
ix) 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.
x) x²+y²+z²=xyz is the equation of a cone.

Show that x = y = z+1 is a secant line of the sphere x² +y² +z²−x−y+z−1 = 0. Also find the intercept made by the sphere on the line.

What is the new equation of the conic x
2 +y
2 +4x−2y+3 = 0, when
i) the origin is shifted at (2,−1), followed by a rotation of axes through 45◦
?
ii) the axes are rotated through 45◦
, followed by the shifting of the origin at
(2,−1)?
Are the equations in i) and ii) above the same? Why?

Find the equation of the plane passing through the line of intersection of the planes
2x+3y+z = 4 and x+y+z = 2, and which is perpendicular to the plane
2x+3y−z = 3.