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The abscissa and ordinate of the end points $\mathrm { A }$ and $\mathrm { B }$ of a focal chord of the parabola $\mathrm { y^2=4 x }$ are respectively the roots of $\mathrm { x^2-3 x+a=0 }$ and $\mathrm { y^2+6 y+b=0 }$ . The equation of the circle with $\mathrm { AB }$ as diameter is

Option: 1
$\mathrm { x^2+y^2-3 x+6 y+3=0}$

Option: 2
$\mathrm { x^2+y^2-3 x+6 y-3=0}$

Option: 3
$\mathrm { x^2+y^2-3 x+6 y-2=0}$

Option: 4
$\mathrm { x^2+y^2-3 x-6 y-3=0}$

Answers (1)

best_answer

$\mathrm {t_1 t_2=-1}$ as $\mathrm {A B}$ is focal chord.

$\begin{gathered} \mathrm { x^2-3 x+a=0 ; x_1+x_2=3 \text { and } x_1 x_2=a }\\ \mathrm {y^2+6 y+b=0 ; y_1+y_2=-6 \text { and } y_1 y_2=b} \\ \mathrm {x_1 x_2=\frac{1}{t_1^2} t_1^2=1=a} \end{gathered}$
$\begin{aligned} & \mathrm {y_1 y_2=2 t_1\left(-\frac{2}{t_1}\right)=-4=b }\\ & \mathrm {a=1, b=-4} \end{aligned}$
Equation of circle $\mathrm{AB}$ is diameter
$\mathrm{ x^2+y^2-3 x+6 y-3=0 \text {. } }$

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Deependra Verma

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