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.
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Expert's answer
2020-03-02T17:14:17-0500
a) A parabola is the set of points P whose distances from a fixed point F in the plane are equal to their distances from a fixed line l in the plane. The fixed point F is called focus and the fixed line l the directrix of the parabola.
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 is
x+2z+2=0
c)
i)The shifted origin has the coordinates (2, -1). Let the coordinates of any point (x,y) on the given line changes to (x',y') on shifting the origin to (2,-1).
The equations in i) and ii) above are not the same.The order of transformations is important.
d) Write the equations in parametric form.
L1:x=2t,y=t+1,z=t−1L2:x=3s−4,y=2s,z=3s+1
The lines are not parallel since the vectors v1=(2,1,1) and v2=(3,2,3) are not parallel. Next we try to find intersection point by equating x,y, and z.
2t=3s−4t+1=2st−1=3s+1
2t−t−1−t+1=3s−4−2s0=s−4s=4
2t=12−4t=8−1t=12+2
We have contradiction. So there is no solution for s and t. Since the two lines are neither parallel nor intersecting, they are skew lines. No plane passes through the lines x+4/3 =y/2 =z−1/3
and x/2 =y−1/1 =z+1/1.
e) Find the equation of the right circular cone whose vertex is (1,2,3), 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
Let P(x,y,z) be any point on the cone with vertex at V(1,2,3) and the axis
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