Question

The equations of the tangents to circle $5x^2+5y^2=1,$ parallel to line $3x+4y=1$ are

• A. $3x+4y=\pm 2\sqrt{5}$
• B. $6x+8y=\pm \sqrt{5}$
• C. $3x+4y=\pm \sqrt{5}$
• D. None of the above

Answer 1

Answer: (C)

Given equation of circle is $5x^2+5y^2=1$
or $x^2+y^2=\frac{1}{5}$
centre of the circle is $(0,0)$.
Equation of tangent which are parallel to
$3x+4y-1=0$ is $3x+4y+\lambda =0$..(i)
We know that perpendicular distance from centre
$(0,0)$ to $3x+4y+\lambda =0$
should be equal to radius.
$\therefore \frac{3\times 0+4\times 0+\lambda }{\sqrt{(3{)}^2+(4{)}^2}}=\pm \frac{1}{\sqrt{5}}$
$\Rightarrow \lambda =\pm \frac{5}{\sqrt{5}}=\pm \sqrt{5}$
On putting the value of $\lambda$ in Eq. (i),
we get $3x+4y\pm \sqrt{5}=0$
or $3x+4y=\pm \sqrt{5}$