A tangent is drawn to the ellipse to cut the ellipse a2x2+b2y2=1 and to cut the ellipse c2x2+d2y2=1 at the points P and Q. If the tangents are at right angles, then the value of (c2a2)+(d2b2) is
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Two circles with radii aandb
touch each other externally such that θ
is the angle between the direct common tangents, (a>b≥2)
. Then prove that θ=2sin−1(a+ba−b)
The line 3x+6y=k
intersects the curve 2x2+3y2=1
at points AandB
. The circle on AB
as diameter passes through the origin. Then the value of k2
If two distinct chords, drawn from the point (p, q) on the circle x2+y2=px+qy
are bisected by the x-axis, then
Statement 1 : The equation of chord through the point (−2,4)
which is farthest from the center of the circle x2+y2−6x+10y−9=0
Statement 1 : In notations, the equation of such chord of the circle S=0
bisected at (x1,y1)
must be T=S˙
Prove that quadrilateral ABCD
, where AB≡x+y−10,BC≡x−7y+50=0,CD≡22x−4y+125=0,andDA≡2x−4y−5=0,
is concyclic. Also find the equation of the circumcircle of ABCD˙
If the join of (x1,y1)
makes on obtuse angle at (x3,y3),
then prove than (x3−x1)(x3−x2)+(y3−y1)(y3−y2)<0
Prove that the equation of any tangent to the circle x2+y2−2x+4y−4=0
is of the form y=m(x−1)+31+m2−2.
Find the length of the chord x2+y2−4y=0
along the line x+y=1.
Also find the angle that the chord subtends at the circumference of the larger segment.