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Let a circle be given by `2x(x-1)+y(2y-b)=0,(a!=0,b!=0)`. Find the condition on `aa n db`if two chords each bisected by the x-axis, can be drawn to the circlefrom `(a , b/2)` |
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Answer» The given circle can be rewritten as `x^(2)+y^(2)-ax-(by)/(2)=0" "...(i)` Let one of the chord through (a, b/2) be bisected at (h, 0). Then, the equation of the chord having (h, 0) as mid-point is 0 `T=S_(1)` `rArr h*x+0*y-(a)/(2)(x+h)-(b)/(h)(y+0)=h^(2)+0-ah-0` `rArr (h-(a)/(2))x-(by)/(4)-(a)/(2)h=h^(2)-ah" "...(ii)` It passes through `(a, b//2)`,then `(h-(a)/(2))a-(b)/(4)*(b)/(2)-(a)/(2)=h^(2)-ah` According to the given condition, Eq. (iii) must have tow distinct real roots. This is possible, if the discriminant of Eq. (iii) is greater than 0. i.e. `(9)/(4)a^(2)-4((a^(2))/(2)+b^(2)/8)gt0rArr (a^(2))/(4)-(b^(2))/(2)gt0` `rArr a^(2)gt2b^(2)` |
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