Pole, Polar and Conjugate Elements - One Line Questions
1.
What is the pole of the line Ax + By + C = 0 with respect to the circle x^2 + y^2 = a^2? —
(-a^2A/C, -a^2B/C)
2.
What is the pole of the line Ax + By + C = 0 with respect to the ellipse (x^2/a^2) + (y^2/b^2) = 1? —
(-a^2A/C, -b^2B/C)
3.
What is the pole of the line Ax + By + C = 0 with respect to the hyperbola (x^2/a^2) - (y^2/b^2) = 1? —
(a^2A/C, -b^2B/C)
4.
What is the pole of the line Ax + By + C = 0 with respect to the parabola y^2 = 4ax? —
(-aB^2/AC, 2aB/A)
5.
Consider the conic Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0. If the polar of (x1, y1) is L, then the pole of L is: —
(x1, y1)
6.
What is the condition for two points (x1, y1) and (x2, y2) to be conjugate with respect to the hyperbola (x^2/a^2) - (y^2/b^2) = 1? —
(x1x2/a^2) - (y1y2/b^2) = 0
7.
What is the condition for two points (x1, y1) and (x2, y2) to be conjugate with respect to the ellipse (x^2/a^2) + (y^2/b^2) = 1? —
(x1x2/a^2) + (y1y2/b^2) = 0
8.
What is the equation of the polar of a point (x1, y1) with respect to the hyperbola (x^2/a^2) - (y^2/b^2) = 1? —
(xx1/a^2) - (yy1/b^2) = 1
9.
What is the equation of the polar of a point (x1, y1) with respect to the ellipse (x^2/a^2) + (y^2/b^2) = 1? —
(xx1/a^2) + (yy1/b^2) = 1
10.
What is the polar of the origin (0,0) with respect to the general conic Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0? —
2Gx + 2Fy + C = 0
11.
What is the condition for the line lx + my + n = 0 to be a tangent to the hyperbola (x^2/a^2) - (y^2/b^2) = 1? —
a^2l^2 - b^2m^2 = n^2
12.
What is the condition for the line lx + my + n = 0 to be a tangent to the ellipse (x^2/a^2) + (y^2/b^2) = 1? —
a^2l^2 + b^2m^2 = n^2
13.
What is the condition for the line lx + my + n = 0 to be a tangent to the parabola y^2 = 4ax? —
a*l^2 = m*n
14.
Two chords of a conic passing through a point P are divided harmonically by P and the polar of P. This is known as the theorem of: —
Harmonic conjugates
15.
If the polars of P and Q with respect to a conic are the same line L, then P and Q are: —
The same point
16.
The locus of the intersection of perpendicular tangents to a conic is called: —
Director circle
17.
In the context of conic sections, what is the eccentricity 'e' for an ellipse? —
0 < e < 1
18.
In the context of conic sections, what is the eccentricity 'e' for a parabola? —
e = 1
19.
In the context of conic sections, what is the eccentricity 'e' for a hyperbola? —
e > 1
20.
What is the eccentricity 'e' for a circle? —
e = 0
21.
What is the locus of points the product of whose distances from two fixed points is constant? —
Hyperbola
22.
What is the locus of points whose distance from a fixed point (focus) is proportional to its distance from a fixed line (directrix)? —
Parabola
23.
What is the locus of points equidistant from a fixed point and a fixed line? —
Parabola
24.
Two lines L1 and L2 are said to be conjugate with respect to a conic if each passes through the pole of the other. If L1 is y = mx + c, what is the condition for L2 to be conjugate to L1? —
The pole of L1 lies on L2.
25.
What is the condition for the lines y = m1x + c1 and y = m2x + c2 to be conjugate with respect to the circle x^2 + y^2 = a^2? —
m1 + m2 = 0
26.
What is the condition for the lines y = m1x + c1 and y = m2x + c2 to be conjugate with respect to the parabola y^2 = 4ax? —
am1m2 = c1 + c2
27.
What is the condition for the lines y = m1x + c1 and y = m2x + c2 to be conjugate with respect to the ellipse (x^2/a^2) + (y^2/b^2) = 1? —
b^2 + m1m2a^2 = 0
28.
What is the condition for the lines y = m1x + c1 and y = m2x + c2 to be conjugate with respect to the hyperbola (x^2/a^2) - (y^2/b^2) = 1? —
b^2 - m1m2a^2 = 0
29.
The tangents from an external point to a conic touch the conic at points whose join is the: —
Polar of the external point
30.
If the tangents at points P and Q on a conic intersect at T, then the polar of T is the line joining which points? —
The points P and Q
31.
If the polar of a point P with respect to a conic passes through a point Q, then the polar of Q passes through which point? —
The point P
32.
Two points P and Q are said to be conjugate with respect to a conic if the polar of P passes through which point? —
The point Q
33.
If the polars of P and Q with respect to a central conic intersect at O, then O is: —
The pole of PQ
34.
The locus of the poles of a system of parallel chords of a conic is a line passing through: —
The center of the conic
35.
What is the director curve of the parabola y^2 = 4ax? —
The directrix
36.
The pole of the directrix of a conic section is located at: —
The focus
37.
What is the definition of the pole of a point with respect to a conic section? —
The line such that the given point is the pole of this line with respect to the conic.
38.
What is the definition of the polar of a point with respect to a conic section? —
The line such that the given point is the pole of this line with respect to the conic.
39.
If P and Q are conjugate points with respect to a central conic, and the line PQ intersects the conic at A and B, then the tangents at A and B intersect on: —
The polar of Q
40.
If P and Q are conjugate points with respect to a conic, then which of the following is true? —
The polar of Q passes through P.
41.
What is the pole of the line Ax + By + C = 0 with respect to the general conic Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0? —
The solution of the system of equations: Ax + Hy + G = -n(Al+Hm+G)/C
42.
The polar of a point on a conic is: —
The tangent at that point
43.
When is the polar of a point P with respect to a conic a tangent to the conic? —
When P lies on the conic.
44.
What is the director circle of the ellipse (x^2/a^2) + (y^2/b^2) = 1? —
x^2 + y^2 = a^2 + b^2
45.
What is the director circle of the hyperbola (x^2/a^2) - (y^2/b^2) = 1? —
x^2 + y^2 = a^2 - b^2
46.
What is the condition for two points (x1, y1) and (x2, y2) to be conjugate with respect to the circle x^2 + y^2 = a^2? —
x1x2 + y1y2 = a^2
47.
What is the equation of the polar of a point (x1, y1) with respect to the circle x^2 + y^2 = a^2? —
xx1 + yy1 = a^2
48.
What is the condition for two points (x1, y1) and (x2, y2) to be conjugate with respect to the parabola y^2 = 4ax? —
y1y2 = 2a(x1 + x2)
49.
If the polars of two points P and Q with respect to a conic are perpendicular, are P and Q necessarily conjugate? —
No
50.
What is the equation of the polar of a point (x1, y1) with respect to the parabola y^2 = 4ax? —
yy1 = 2a(x + x1)