Home
Class 12
MATHS
Find the locus of point P such that the ...

Find the locus of point `P` such that the tangents drawn from it to the given ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` meet the coordinate axes at concyclic points.

Text Solution

Verified by Experts

Let `P-=(h,k)`. The combined equation of tangesnts from (h,k) is
`((x^(2))/(a^(2))+(y^(2))/(b^(2)))((h^(2))/(a^(2))+(k^(2))/(b^(2))-1)-((hc)/(a^(2))+(yk)/(b^(2))-1)^(2)=0`
Any coin that can be drawn through the points of intersection of these tangents witht eh coordinate axes is
`((x^(2))/(a^(2))+(y^(2))/(b^(2)))((h^(2))/(a^(2))+(k^(2))/(b^(2))-1)-((hc)/(a^(2))+(yk)/(b^(2))-1)^(2)+lambdaxy=0`
`x^(2)((k^(2))/a^(2)b^(2)-(1)/(d^(2)))+y^(2)((h^(2))/(a^(2)b^(2))-(1)/(b^(2)))+xy(lambda-(2hk)/(a^(2)b^(2)))+(2xh)/(a^(2))+(2yk)/(b^(2))-(h^(2))/(a^(2))-(k^(2))/(b^(2))=0`
It should represent a circle. Therefore,
`lambda=(2hk)/(a^(2)b^(2)')(k^(2))/(a^(2)b^(2))-(1)/(a^(2))=(h^(2))/(a^(2)b^(2))-(1)/(b^(2))`
`:. lambda=(2hk)/(a^(2)b^(2)),h^(2)-k^(2)=a^(2)-b^(2)`
Hence, the locus of P is
`x^(2)-y^(2)=a^(2)-b^(2)`
Promotional Banner

Similar Questions

Explore conceptually related problems

The locus of the point which is such that the chord of contact of tangents drawn from it to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 forms a triangle of constant area with the coordinate axes is (a) straight line (b) a hyperbola (c) an ellipse (d) a circle

The locus of the middle point of the intercept of the tangent drawn from an external point to the ellipse x^(2) + 2y^(2) =2 between the coordinate axes is-

Find the locus of the foot of the perpendicular drawn from the center upon any tangent to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1.

Find the equations of the tangents drawn from the point (2, 3) to the ellipse 9x^2+16 y^2=144.

Find the locus of the midpoints of the all chords drawn from the ends points of the minor axis of an ellipse x^2/a^2+y^2/b^2=1

The locus of the point which divides the double ordinates of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 in the ratio 1:2 internally is

Find the lengths of the tangents drawn from the point. (x_(1),y_(1)) to the circle x^(2)+y^(2)+2x=0

Find the lengths of the tangents drawn from the point. (-1,1) to the circle x^(2)+y^(2)-2x+4y+1=0

Find the lengths of the tangent drawn from the point. (2,-2) to the circle 3(x^(2)+y^(2))-4x-7y=3

Find the lengths of the tangents drawn from the point. (-4,5) to the circle x^(2)+y^(2)=16