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  • The abscissa and ordinates of the points A and B are the roots of the equations <img alt="x^2+2 a x+b=0 \text { and } x^2+2 c x+d=0" src="https://entrancecorner.oncodecogs.com/gif.latex?x%5E2&p

The abscissa and ordinates of the points A and B are the roots of the equations x^2+2 a x+b=0 \text { and } x^2+2 c x+d=0 respectively.

Equation of circle with AB as diameter is

Option: 1

x^2 + y^2 + 2ax + 2cy + b + d = 0


Option: 2

x^2+y^2-2 a x-2 c y-b-d=0


Option: 3

x^2+y^2-2 a x-2 c y+b+d=0


Option: 4

x^2+y^2+2 a x+2 c y-b-d=0


Answers (1)

best_answer

If roots of \mathrm{x^2+2 a x+b=0} are x_1 and x_2 then \mathrm{x_1+x_2=-2 a \: and \: x_1 x_2=b}
Similarly if roots of \mathrm{x^2+2 c x+d=0 \: are \: y_1 \: and\: y_2} then

\mathrm{y}_1+\mathrm{y}_2=-2 \mathrm{c}, \mathrm{y}_1 \mathrm{y}_2=\mathrm{d}
Now, the equation of the required circle is 

\begin{aligned} & \left(\mathrm{x}-\mathrm{x}_1\right)\left(\mathrm{x}-\mathrm{x}_2\right)+\left(\mathrm{y}-\mathrm{y}_1\right)\left(\mathrm{y}-\mathrm{y}_2\right)=0 \\ & \mathrm{x}^2+\mathrm{y}^2-\mathrm{x}\left(\mathrm{x}_1+\mathrm{x}_2\right)-\mathrm{y}\left(\mathrm{y}_1+\mathrm{y}_2\right)+\mathrm{x}_1 \mathrm{x}_2+\mathrm{y}_1 \mathrm{y}_2=0 \end{aligned}
\mathrm{\Rightarrow x^2+y^2+2 a x+2 c y+b+d=0}
 

 

Posted by

rishi.raj

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