Question

Line $3x-4y=k$ will cut the circle $x^2+y^2-2x+4y-11=0$ at distinct points, if

• A. $k>\frac{25}{7}$
• B. $15<k<30$
• C. $-9<k<31$
• D. None of these

Answer 1

Answer: (C)

$-9<k<31$

Centre of the circle is $C(1,-2)$ and radius is $\sqrt{1+4+11}=4$
Now for line $3x-4y=k$ to cut the circle at distinct point.
Distance of line from centre of the circle should be less than its radius.
$\Rightarrow \left|\frac{3\left(1\right)-4(-2)-k}{5}\right|<4$
$\Rightarrow |k-11|<20\Rightarrow -20<k-11<20\Rightarrow -9<k<31$