Question

Let $ax+by+c=0$ be a variable straight line, where a, b and c are $1$st, $3$rd and $7$th terms of some increasing A.P. Then the variable straight line always passes through a fixed point which lies on

• A. $x^2+y^2=13$
• B. $x^2+y^2=5$
• C. $y^2=9x$
• D. $3x+4y=9$

Answer 1

The correct option is A $x^2+y^2=13$

Given equation

$ax+by+c=0-----\left(1\right)$

Here $a,b,c$ are $1st,3rd,7th$ terms of increasing $A.P$

Hence $b=a+2$ and $c=a+6$

From equation (1)

$ax+(a+2)y+(a+6)=0$

Here on putting $x=2,y=-3$ the above equation is satisfied

Hence fixed point is $(2,3)$

Here one point is fixed and other point is variable So circle is formed

Radius of circle is equal to distance between $(0,0)$ and $(2,3)$

$r=\sqrt{2^2+3^2}=\sqrt{13}$

Eqaution of circle with centre $(0,0)$ and radius $r=\sqrt{13}$

$x^2+y^2=(\sqrt{13}{)}^2$

$x^2+y^2=13$