Question

The length of the chord cut off by $y=2x+1$ from the circle $x^2+y^2=2$ is

• A. $\frac{5}{6}$
• B. $\frac{6}{5}$
• C. $\frac{6}{\sqrt{5}}$
• D. $\frac{3}{\sqrt{5}}$

Answer 1

Answer: (C)

$\frac{6}{\sqrt{5}}$

$y=2x+1$ is a chord of circle $x^2+y^2=2$
$x^2+{(2x+1)}^2=2$
$x^2+{4x}^2+1+4x=2$
${5x}^2+4x-1=0$
${5x}^2+5x-x-1=0$
$5x(x+1)-1(x+1)=0$
$(x+1)(5x-1)=0$
$x=-1,+\frac{1}{5}$
Put values of x in $y=2x+1$
$x=-1;y=-2+1=-1⟹(-1,-1)$
$x=\left(\frac{1}{5}\right);y=\frac{2}{5}+1=\frac{7}{5}⟹(\frac{1}{5},\frac{7}{5})$
$\therefore$ chord intersects circle at $(-1,-1)$ & $(\frac{1}{5},\frac{7}{5})$
$\therefore$ length of chord = Distance between $(-1,-1)$ & $(\frac{1}{5},\frac{7}{5})$
$=\sqrt{{(\frac{1}{5}+1)}^2+{(\frac{7}{5}+1)}^2}$
$=\sqrt{{\left(\frac{6}{5}\right)}^2+{\left(\frac{12}{5}\right)}^2}$
$=\sqrt{\frac{36}{25}+\frac{144}{25}}$$=\sqrt{\frac{180}{25}}$
$=\sqrt{\frac{36}{5}}$
$=\frac{6}{\sqrt{5}}$