Question

If the line x + y = 5 is a tangent to the circle $x^2+y^2-2x-4y+3=0$, then the coordinates of the point of contact are .

• A. (2,3)
• B. (-2,-3)
• C. (-2,3)
• D. (2,-3)

Answer 1

The correct option is A (2,3)

Let the coordinates of the point of contact be (x', y').
$\therefore$ the equation of the tangent to the given circle at the point (x',y') is:
xx' + yy' - (x + x') - 2(y + y') + 3 = 0
$⟹$ (x' - 1)x + (y' - 2)y = x' + 2y' - 3 ...(i)
The equation of the given line is x + y = 5 ...(ii)
From (i) and (ii):
$\frac{x\text{'}-1}{1}=\frac{y\text{'}-2}{1}=\frac{x\text{'}+2y\text{'}-3}{5}=\lambda \Rightarrow x\text{'}=\lambda +1,y\text{'}=\lambda +2\mathrm{and}x\text{'}+2y\text{'}-3=5\lambda \Rightarrow (\lambda +1)+2(\lambda +2)-3=5\lambda \Rightarrow \lambda =1\Rightarrow x\text{'}=2\mathrm{and}y\text{'}=3$