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Definition of Circle

The abscissa ...

Question

The abscissa of two points A and B are the roots of the equation $x^{2} + 2 a x - b^{2}$ and their ordinates are the root of the equation $x^{2} + 2 p x - q^{2} = 0$ . the equation of the circle with AB as diameter is

x^{2} + y^{2} + 2 a x + 2 p y + b^{2} + q^{2} = 0

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x^{2} + y^{2} - 2 a x - 2 p y - b^{2} - q^{2} = 0

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x^{2} + y^{2} + 2 a x + 2 p y - b^{2} - q^{2} = 0

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Solution

The correct option is D $x^{2} + y^{2} + 2 a x + 2 p y - b^{2} - q^{2} = 0$
$x^{2} + 2 a x - b^{2} = 0$

Let roots be $x_{1}$ & $x_{2}$

$x_{1} + x_{2} = \frac{- b}{a} = - 2 a$
$x_{1} x_{2} = \frac{c}{a} = - b^{2}$
$x^{2} + 2 p x - q^{2} = 0$

Let roots be $y_{1}$ & $y_{2}$

$y_{1} + y_{2} = - 2 p$
$y_{1} y_{2} = - q^{2}$

Equation in diametric form is
$x^{2} + y^{2} - (x_{1} + x_{2}) x - (y_{1} + y_{2}) y + x_{1} x_{2} + y_{1} y_{2} = 0$
$x^{2} + y^{2} + 2 a x + 2 p y - b^{2} - q^{2} = 0$

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The abscissae of A and B are the roots of the equation $x^{2} + 2 a x - b^{2} = 0$ and their coordinates are the roots of the equation $y^{2} + 2 p y - q^{2}$ = 0. The equation of the circle with AB as diameter

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x^{2} + 2 a x - b^{2} = 0

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y^{2} + 2 p y - q^{2} = 0

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and

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Q. The abscissae of the two points A and B are the roots of the equation x² + 2ax − b² = 0 and their ordinates are the roots of the equation x² + 2px − q² = 0. Find the equation of the circle with AB as diameter. Also, find its radius.

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The abscissa of two points A and B are the roots of the equation x2+2ax−b2 and their ordinates are the root of the equation x2+2px−q2=0. the equation of the circle with AB as diameter is

The correct option is D x2+y2+2ax+2py−b2−q2=0x2+2ax−b2=0Let roots be x1 & x2x1+x2=−ba=−2ax1x2=ca=−b2x2+2px−q2=0Let roots be y1 & y2y1+y2=−2py1y2=−q2Equation in diametric form is x2+y2−(x1+x2)x−(y1+y2)y+x1x2+y1y2=0x2+y2+2ax+2py−b2−q2=0

The abscissa of two points A and B are the roots of the equation $x^{2} + 2 a x - b^{2}$ and their ordinates are the root of the equation $x^{2} + 2 p x - q^{2} = 0$ . the equation of the circle with AB as diameter is