Analytic Geometry Answers

SHOW THAT THE PLANE AX+BY+CZ+D=0 TOUCHES THE 3X^2-6Y^2+9Z^2+17=0 FIND THE POINT OF CONTACT

Full notes on vector concept

If a=10, b =13 and the angle between a and b is 59.5 degree, what is the scalar product of a and b

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.