Question

If the tangent at the point P on the circle $x^2+y^2+6x+6y=2$ meets the straight line $5x-2y+6=0$ at a point Q on the y-axis then the length of PQ is

• A. $4$
• B. $2\sqrt{5}$
• C. $5$
• D. $3\sqrt{5}$
• E. $6$

Answer 1

The correct option is C $5$

circle equation : $x^2+y^2+6x+6y=2;C_1$

equation of line: $5x-2y+6=0:L_1$

tangent at point $P$ on $C_1$ meet $L_1$ at point $d$

As tangent intersect $y-$axis at $Q,\therefore x=0$

$5\left(0\right)-2y+6=0$

$y=3$

coordinate of $Q=(0,3)$

$f(x,y)=x^2+y^2++6x+6y=2$

length of tangent

$PQ=\sqrt{f(0,3)}$

$=\sqrt{0^2+3^2+6.0+6.3-2}$

$=\sqrt{25}$

$=5$

length of tangent $PQ=5cm$

option $C$