Analytic Geometry Answers

Find the angle between the lines whose direction cosines satisfy the equations l+m+n=0 and 2nl+2lm-mn=0.

Which coordinate notation represent the translation of P(-2, 5) to P’ (9, -12)?

P(4,9) ,Q(3,2),R(13,-3) and S(11,13) are four points in a plane and L is the mid point of RS . the line through Q perpendicular to RS meets PR at M find :
1 the equation of the lines QL and PR
2 the coordinates of the point of intersection N ,of QL and PR
3 the length of QM
4 the area of triangle PQN
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P is a point on the plane lx + my + nz = p and a point Q is taken on the line such that OP. OQ= p^2 ; show that the locus of point Q is p(lx + my + nz)= x^2 + y^2 + z^2.

The coordinates of the points A and B in the x-y plane are (-3,-4) and (5,0) respectively.
(a) Obtain the equation of the circle on AB as diameter, and show that the point (-3,0) lies on the circle.
(b) Find also:
(i) the equation of a line CD in parallel to the line 2x +2y-7=0 whose intercept on the x-axis is -3
(ii) the coordinates of the points of intersection of the line CD with the circle
(iii) the equation of the tangent of the circle at the point (-3,0)

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

Which of the following statements are true and which are false? Give reasons for your
answer.
i) The equation r = acos(θ +α) +bsin(θ +α) represents a circle.
ii) The direction ratios of the line x−5 = 5−y,z = 5 are 1,1,5.
iii) The intersection of a plane and a cone can be a pair of lines.
iv) The angle between the planes x+2y+2z = 5 and 2x+2y+3 = 0 is 60°.
v) The equations 2x^2 +y^2 +3z^2 + 4x + 4y + 18z + 34 = 0, 2x^2 − y^2 = 4y − 4y − 4x represent a real conic.
vi) 4x^2 − 9y^2 + z^2 + 36 = 0 represents a hyperboloid of one sheet.
vii) The intersection of any plane with an ellipsoid is an ellipse.
viii) No plane passes through the points (1,2,3),(1,−1,0) and (1,1,2).
ix) The circle with centre (a,0) and radius a, where a > 0, touches all the sides of the square x = 0, x = a, y = ±a.
x) If the projection of a line segment AB on a line L is 0, then AB lies in L.

Which of the following statements are true and which are false? Please give reasons for the answers.
1.) 4x²-9y²+z²+36=0 represents a hyperboloid of one sheet.
2.) The intersection of any plane with an ellipsoid is an ellipse.
3.) No plane passes through the points (1,2,3), (1,-1,0) and (1,1,2).
4.) The circle with centre (a,0) and radius a, where a>0, touches all the sides of the square x = 0,x =a,y=±a.
5.) If the projection of a line segment AB on a line L is 0, then AB lies in L.

Which of the following statements are true and which are false? Please give reasons for the answers.
1.) The equation r=acos(θ+α)+bsin(θ+α) represents a circle.
2.) The direction ratios of the line x-5=5-y, z=5 are 1,1,5.
3.) The intersection of a plane and a cone can be a pair of lines.
4.) The angle between the planes x+2y+2z=5 and 2x+2y+3=0 is 60º
5.) The equations 2x²+y²+3z²+4x+4y+18z+34=0, 2x²-y²=4y-4y-4x represent a real conic.

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