If the line y=2x+c is a tangent to the circle | vedclass

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(A) The given line is $y=2x+c$,which can be written as $2x-y+c=0$. The equation of the circle is $x^2+y^2=5$,which has center $(0,0)$ and radius $r=\s

Vedclass · Accepted Answer

$5$

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(A) The given line is $y=2x+c$,which can be written as $2x-y+c=0$. The equation of the circle is $x^2+y^2=5$,which has center $(0,0)$ and radius $r=\sqrt{5}$. For a line $Ax+By+C=0$ to be tangent to a circle with center $(h,k)$ and radius $r$,the perpendicular distance from the center to the line must be equal to the radius: $\frac{|A(h)+B(k)+C|}{\sqrt{A^2+B^2}} = r$ Substituting the values $A=2, B=-1, C=c, h=0, k=0, r=\sqrt{5}$: $\frac{|2(0)-1(0)+c|}{\sqrt{2^2+(-1)^2}} = \sqrt{5}$ $\frac{|c|}{\sqrt{4+1}} = \sqrt{5}$ $\frac{|c|}{\sqrt{5}} = \sqrt{5}$ $|c| = \sqrt{5} 	imes \sqrt{5} = 5$ Therefore,$c = \pm 5$. Comparing with the given options,the value of $c$ is $5$.

If the line $y=2x+c$ is a tangent to the circle $x^2+y^2=5$,then a value of $c$ is

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