Question

A circle touches the $y$-axis at the point $(0,4)$ and passes through the point $(2,0)$. Which of the following lines is not a tangent to the circle?

• A. $4x-3y+17=0$
• B. $3x+4y-6=0$
• C. $4x+3y-8=0$
• D. $3x-4y-24=0$

Answer 1

The correct option is C $4x+3y-8=0$

Given circle is


We know that,
$AM=r-2AC=r$
Now, $A{C}^2=C{M}^2+A{M}^2$
$\Rightarrow r^2=16+(r-2{)}^2\Rightarrow r^2=16+r^2-4r+4\Rightarrow r=5$
Therefore, the centre is $(5,4)$
Distance from centre to the tangent is equal to radius.

For $4x-3y+17=0$,
$\frac{|20-12+17|}{5}=5$

For $3x+4y-6=0$,
$\frac{|15+16-6|}{5}=5$

For $4x+3y-8=0$,
$\frac{|20+12-8|}{5}\ne 5$

For $3x-4y-24=0$,
$\frac{|15-16-24|}{5}=5$


Alternate solution :
$O{D}^2=OA\times OB\Rightarrow 16=2\times OB\Rightarrow OB=8\therefore AB=6\therefore AM=3,CM=4\Rightarrow CA=5\therefore OM=5$

Centre will be $(5,4)$ and radius is $5$.
Now, checking for all the options
Option (c) is not a tangent,
$\because$ for $4x+3y-8=0$
$\frac{20+12-8}{\sqrt{3^2+4^2}}=\frac{24}{5}\ne 5$