Question

The equation of tangent to a circle $C_1$ centred at the origin is $3x+4y=5$. What will be the equation of the circle $C_2$ concentric to given circle , given that the tangent to $C_1$ intercept a length $8$ units on $C_2$.

• A. $x^2+y^2=16$
• B. $x^2+y^2=19$
• C. $x^2+y^2=15$
• D. $x^2+y^2=17$

Answer 1

The correct option is D $x^2+y^2=17$

Length of intercept $=AB=8$
$\Rightarrow AC=4$
$OC=$ Length of perpendicular from $(0,0)$ to $3x+4y=5$
$\Rightarrow OC=\frac{|3\left(0\right)+4\left(0\right)-5|}{\sqrt{3^2+4^2}}=1$
Now
${OA}^2={OC}^2+{AC}^2{OA}^2=1+16=17$
Radius of $C_2=OA$ and centre is $(0,0)$ as the circles are concentric
So the equation of $C_2$ is
$x^2+y^2={OA}^2\Rightarrow x^2+y^2=17$
So option $D$ is correct.