I: The locus of poles of chords of the ellipse x2/a2+y2/b2=1 which touch the parabola y2= 4px is p a2y2+b4x=0 II: The locus of poles of chords of the ellipse x2/a2+y2/b2=1 which touch the x2/α2+y2/β2=1 is α 2x2/a4+β2y2/b4=1
A. Only I is true
B. Only II is true
C. both I and II are true
D. neither I nor II true
The locus of the middle point of chords of the ellipse x2/a2+y2/b2=1 which are at a constant distance d from the centre of the ellipse is
A. (x2/a2+y2/b2)2=d2 (x2/a4+y2/b4)
B. (x2/a2+y2/b2)2= (x2/a4+y2/b4)
C. (x2/a2+y2/b2)2=2d2 (x2/a4+y2/b4)
D. (x2/a2+y2/b2)2=d2 (x2/a4-y2/b4)
The locus of midpoint of the chord of the ellipse x2/a2+y2/b2=1 which pass through the fixed point (h, k) is
A. x2/a2+y2/b2=xh/a2+yk/b2
B. x2/a2+y2/b2=xh/a2-yk/b2
C. x2/a2-y2/b2=xh/a2+yk/b2
D. x2/a2-y2/b2=xh/a2-yk/b2
The equation of the auxiliary circle of x2/12+y2/18=1 is
A. x2+y2=12
B. x2+y2=18
C. x2+y2=6
D. x2+y2=30
If the chord of contact of the point (1, -2) with respect to the ellipse 4x2+5y2=20 is ax+by+c=0 then the ascending order of a, b, c is
A. a, b, c
B. c, b, a
C. c, a, b
D. b, a, c
The locus of a point P for which the chord of contact of x2/a2+y2/b2=1 touch the circle x2+y2=r2
A. x2/a4+y2/b4=1/r4
B. x2/a4+y2/b4=1/r2
C. y2/a4+x2/b4=1/r2
D. x2/a4+y2/b4=-1/r2
The value of k if (1, 2) (k, -1) are conjugate points with respect to the ellipse 2x2+3y2=6 is
A. 2
B. 4
C. 6
D. 8
Pole of the line 2x+3y+4=0 w.r.to the ellipse x2/2+y2/4=1 is
A. (1, 3)
B. (1, -3)
C. (-1, 3)
D. (-1, -3)
The equation of the normal at a point hose eccentric angle is 3π/2+θ to the ellipse x2/9+y2/4=1 is
A. 3x/sin θ+2y/cos θ=5
B. 3x/sin θ-2y/cos θ=5
C. 2x/sin θ+3y/cos θ=5
D. 2x/sin θ-3y/cos θ=5
If 2x-y+3=0, 4x+ky+3=0 are conjugate with respect to the ellipse 5x2+6y2-15=0 then k=
A. 1
B. 2
C. 3
D. 6
P(θ) and D(θ+π/2) are two points on the ellipse x2/a2+y2/b2=1. The locus of point of intersection of tangents at P and D to the ellipse is
A. x2/a2+y2/b2=a
B. x2/a2+y2/b2=b
C. x2/a2+y2/b2=2
D. x2/a2+y2/b2=ab
If the variable line l1(x-a)+y=0 and l2(x+a)+y=0 are conjugate lines w. r. to the ellipse x2/a2+y2/b2=1. Then the locus of their point of intersection is
A. x2/a2+y2/b2=1
B. x2/a2+2y2/b2=1
C. x2/a2+y2/b2=2
D. none
The locus of the poles w.r.t the ellipse x2/a2+y2/b2=1 of tangents to its auxiliary circle is
A. x2/a2+y2/b2=1/a2
B. x2/a2+y2/b2=1/b2
C. x2/a4+y2/b4=1/a2
D. x2/a4+y2/b4=1/b2
If the line lx+my=1 is a normal to the ellipse x2/a2+y2/b2=1 then a2/l2-b2/m2=1
A. a2-b2
B. a2+b2
C. (a2+b2)2
D. (a2-b2)2
The radius of the circle passing through the foci of the ellipse x2/16+y2/9=1 and having its center at (0, 3) is
A. 4
B. 3
C. √12
D. 7/2
If x and y are strictly positive such that x+y =1,then the minimum value of x log x + y log y is
A. log 2
B. - log 2
C. 2 log2
D. 0
The locus of middle point of the chord of the ellipse x2/a2+y2/b2=1 touching the ellipse The locus of midpoint of the chord of the ellipse x2/α2+y2/β2=1
A. α2x2/a4+β2y2/b4=( x2/a2+y2/b2)2
B. α2x2/a4+β2y2/b4=( x2/a2-y2/b2)2
C. α2x2/a4+β2y2/b4=( x2/a2+y2/b2)2
D. α2x2/a4-β2y2/b4=( x2/a2-y2/b2)2
The points on the ellipse x2/25+y2/9=1 whose angles differ by a right angle are
A. (5 cos θ, 3 sin θ), (5 sin θ, 3 cos θ)
B. (5 cos θ, 3 sin θ), (-5 sin θ, 3 cos θ)
C. (5 cos θ, -3 sin θ), (5 sin θ, 3 cos θ)
D. (5 cos θ, -3 sin θ), (-5 sin θ, 3 cos θ)
The chord of contact of tangents drawn from points on x2/a2+y2/b2=1 to the circle x2+ y2=c2 touches the ellipse
A. a2x2-b2y2=c4
B. a2x2+b2y2=c4
C. a2x2+b2y2=c2
D. b2x2+a2y2=c4
The tangents are drawn to the ellipse x2/a2+y2/b2=1 at point where it is intersected by the line lx+my+n=0. The point of intersection of tangents at these points is
A. (a2l/n, b2m/n)
B. (-a2l/n, b2m/n)
C. (a2l/n, -b2m/n)
D. (-a2l/n, -b2m/n)
If the lines 4x+3y-1=0,x-y+5=0 and kx+5y-3=0 are concurrent,then k is equal to
A. 4
B. 5
C. 6
D. 7
The product of the slopes of the tangents to the ellipse 2x2+3y2=6 draw from the point (1, 2) is
A. 1
B. 2
C. -1
D. -2
The ordinate PN of P(a cos θ, b sin θ) on the ellipse x2/a2+y2/b2=1 meets the auxiliary circle at Q,. The locus of the point of intersection of normal at P and Q is
A. x2+y2=(a+b)2
B. x2+y2=(a-b)2
C. x2-y2=(a+b)2
D. x2-y2=(a-b)2
The locus of the middle point of chords of the ellipse x2/a2+y2/b2=1 whose pole lies on the auxiliary circle is
A. (x2/a2+y2/b2)2= (x2+y2)/a2
B. (x2/a2+y2/b2)2= (x2-y2)/a2
C. (x2/a2-y2/b2)2= (x2+y2)/a2
D. (x2/a2-y2/b2)2= (x2-y2)/a2
The midpoint of a chord 4x+5y-13=0 of the ellipse 2x2+5y2=20 is
A. (2, -1)
B. (-2,1)
C. (-2, -1)
D. (2, 1)
The equation of the chord of the ellipse 2x2+3y2=6 having (1, -1) as its midpoint is
A. 8x+9y-25=0
B. 2x-3y-5=0
C. x+y-1=0
D. 3x-2y-6=0
The locus of the poles of chords of the ellipse x2/a2+y2/b2=1 which touch The locus of the poles w.r.t the ellipse x2/α2+y2/β2=1 is
A. α2 x2/a2+ β2 y2/b2=1
B. α2 x2/a4+ β2 y2/b4=1
C. α2 x2/b4+ β2 y2/a4=1
D. α2 x2/b2+ β2 y2/a2=1
The tangent and normal to the ellipse 4x2+9y2 =36 at a point P on it meets the major axis in Q nd R respectively. If QR=4, then the eccentric angle of P is
A. Cos-1 3/5
B. Cos-1 2/3
C. Cos-1 1/3
D. Cos-1 1/5
If the polar of (2, -1) with respect to the ellipse 3x2+4y2=12 is ax+by+c=0 then the ascending order of a, b, c is
A. a, b, c
B. c, b, a
C. c, a, b
D. b, a, c
The total number of real tangents that can be drawn to the ellipse 3x2+5y2=32 and 25x2+9y2=450 passing through (3, 5) is
A. 0
B. 2
C. 3
D. 4