Question

If the line $ax+by=2$ is a normal to the circle $x^2+y^2-4x-4y=0$ and a tangent to the circle $x^2+y^2=1$, then which of the following is true?

• A. $a=\frac{1}{2},b=\frac{1}{2}$
• B. $a=\frac{1+\sqrt{7}}{2},b=\frac{1-\sqrt{7}}{2}$
• C. $a=\frac{1}{4},b=\frac{3}{4}$
• D. $a=1,b=\sqrt{3}$

Answer 1

The correct option is B $a=\frac{1+\sqrt{7}}{2},b=\frac{1-\sqrt{7}}{2}$

Given that $ax+by=2$ is a normal to $x^2+y^2–4x-4y$

The line passes through $(2,2)$ i.e. center of the circle

$⟹2a+2b=2⟹a+b=1-eq.1$

Also given that line is a tangent to $x^2+y^2=1,C=(0,0),r=1$

Perpendicular distance from center to line = radius

$⟹\frac{|0+0–2|}{\sqrt{a^2+b^2}}=1$

$⟹a^2+b^2=4$

$⟹(a+b{)}^2-2ab=4$

$⟹2ab=-3-eq.2$

Solving eq.1 and eq.2

$a=\frac{1+\sqrt{7}}{2},b=\frac{1-\sqrt{7}}{2}$