IIT JEE , AIEEE Forum - Mathematics , Analytical Geometry - analytical geometry-circle

a.z

The locus of the middle point of the chords of the circle of radius r which subtend an angle 45 at any point on the circumference of the circle is a concentric circle with radius equal to

a. r/2

b. 2r/3

c. r/rt2

d. r/rt3

pink_ele

if r is radius of circle
den length of such chord
l=r rt2
now
reuired radius is d distace of mid pt of such a chord frm centre
so radius,
by py thagoras theorm
R=rt(r^2-(r /rt2)^2)
R=r/rt2
option c

allamraju

Assume the centre of circle to be origin and (x₀,y₀) to be the midpoint of the chord.Since it subtends 45⁰ at the circumference,it subtends 90⁰ at the centre.It's eqn. is S₀=S₀₀ where S is x²+y²=r² that is xx₀+yy₀=x₀²+y₀².By homogenisation,

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Assume the centre of circle to be origin and (x₀,y₀) to be the midpoint of the chord.Since it subtends 45⁰ at the circumference,it subtends 90⁰ at the centre.It's eqn. is S₀=S₀₀ where S is x²+y²=r² that is xx₀+yy₀=x₀²+y₀².By homogenisation,

x²+y²=r²[xx₀+yy₀/x₀²+y₀²]².Equating the sum of the coefficients of x² and y² on both sides,we get 2=r²/x₀²+y₀².hence the ans is r/rt2

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Assume the centre of circle to be origin and (x0,y0) to be the midpoint of the chord.Since it subtends 450 at the circumference,it subtends 900 at the centre.It's eqn. is S0=S00 where S is x2+y2=r2 that is xx0+yy0=x02+y02.By homogenisation,

x2+y2=r2[xx0+yy0/x02+y02]2.Equating the sum of the coefficients of x2 and y2 on both sides,we get 2=r2/x02+y02.hence the ans is r/rt2

Assume the centre of circle to be origin and (x₀,y₀) to be the midpoint of the chord.Since it subtends 45⁰ at the circumference,it subtends 90⁰ at the centre.It's eqn. is S₀=S₀₀ where S is x²+y²=r² that is xx₀+yy₀=x₀²+y₀².By homogenisation,

x²+y²=r²[xx₀+yy₀/x₀²+y₀²]².Equating the sum of the coefficients of x² and y² on both sides,we get 2=r²/x₀²+y₀².hence the ans is r/rt2