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If the line $a x+y=c$ , touches both the curves $x^2+y^2=1 \text { and } y^2=4 \sqrt{2} x$ , then |c| is equal to
Option: 1
$\begin{aligned} & \frac{1}{\sqrt{2}} \\ \end{aligned}$

Option: 2
$2 \\$

Option: 3
$\sqrt{2} \\$

Option: 4
$\frac{1}{2}$

Answers (1)

best_answer

Since, equation of given parabola is $y^2=4 \sqrt{2} x$ and

equation of tangent line is , $a x+y=c$

then $c=\frac{\sqrt{2}}{m}=\frac{\sqrt{2}}{-a}$

[ $\because$ line $y=m x+c$ c = + touches the parabola $y^2=4 a x \text { iff } c=a / m$ ]

Then,

equation of tangent line becomes =

$y=-a x-\frac{\sqrt{2}}{a}$ —-------(i)

$\because$ Line (i) is also tangent to the circle $x^2+y^2=1$

$\therefore$ Radius $=1=\frac{\left|-\frac{\sqrt{2}}{a}\right|}{\sqrt{1+a^2}}$

$\Rightarrow \sqrt{a+a^2}=\left|-\frac{\sqrt{2}}{a}\right|$

$\begin{aligned} & \Rightarrow a^4+a^2-2=0 \\ & \Rightarrow\left(a^2+2\right)\left(a^2-1\right)=0 \\ & \Rightarrow a^2=1 \\ & \therefore|c|=\frac{\sqrt{2}}{|a|}=\sqrt{2} \end{aligned}$

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vinayak

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