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If a variable line, $3x+4y-\lambda =0$ is such that the two circles $x^{2}+y^{2}-2x-2y+1=0$ and $x^{2}+y^{2}-18x-2y+78=0$ are on its opposite sides, then the set of all values of $\lambda$ is the interval :
Option 1)
$\left ( 23,31 \right )$
Option 2)
$\left ( 2,17 \right )$
Option 3)
$[12,21]$
Option 4)
$[13,23]$

Answers (1)

best_answer

Perpendicular distance of a point from a line -

$\rho =\frac{\left | ax_{1}+by_{1}+c\right |}{\sqrt{a^{2}+b^{2}}}$

- wherein

$\rho$ is the distance from the line $ax+by+c=0$ .

General form of a circle -

$x^{2}+y^{2}+2gx+2fy+c= 0$

- wherein

centre = $\left ( -g,-f \right )$

radius = $\sqrt{g^{2}+f^{2}-c}$

$3x+4y-\lambda =0$

$(7-\lambda )(31-\lambda )<0$ (Since centre are on the opposite sides)

$=>\lambda \epsilon (7,31)$ ...............(1)

$\left | \frac{7-\lambda }{5} \right |\geq 1\: \: and \: \: \left | \frac{7 -\lambda }{5} \right |\geq 2$

$\left | {7-\lambda } \right |\geq 5\: \: and \: \: \left | {31 -\lambda } \right |\geq 10$

$=>\lambda \leq 2\: \: or\: \: \lambda \geq 12$ ....................................(2)

and

$=>\lambda \leq 21\: \: or\: \: \lambda \geq 41$ .................................(3)

(1) $\cap$ (2) $\cap$ (3)

$\lambda \epsilon [12,21]$

Option 1)

$\left ( 23,31 \right )$

Option 2)

$\left ( 2,17 \right )$

Option 3)
$[12,21]$
Option 4)

$[13,23]$

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