### Eamcet - Maths - Ellipse

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