If the point of intersection of the tangents | vedclass

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(B) The equation of the circle is $x^2 + y^2 - 2x - 6y - 8 = 0$.Let the point of intersection of the tangents be $P(a, b)$. The equation of the chord

Vedclass · Accepted Answer

-$44$

Vedclass · Answer

(B) The equation of the circle is $x^2 + y^2 - 2x - 6y - 8 = 0$.Let the point of intersection of the tangents be $P(a, b)$. The equation of the chord of contact of the tangents drawn from $P(a, b)$ to the circle is given by $T = 0$,which is:$xa + yb - (x + a) - 3(y + b) - 8 = 0$$(a - 1)x + (b - 3)y - (a + 3b + 8) = 0$This chord of contact is the same as the given line $5x + y + 1 = 0$.Comparing the coefficients of the two equations:$\frac{a - 1}{5} = \frac{b - 3}{1} = \frac{-(a + 3b + 8)}{1}$From $\frac{a - 1}{5} = \frac{b - 3}{1}$,we get $a - 1 = 5b - 15 \Rightarrow a - 5b = -14$ $(i)$From $\frac{b - 3}{1} = -(a + 3b + 8)$,we get $b - 3 = -a - 3b - 8 \Rightarrow a + 4b = -5$ (ii)Solving equations $(i)$ and (ii):Subtracting (ii) from $(i)$: $(a - 5b) - (a + 4b) = -14 - (-5)$ $\Rightarrow -9b = -9$ $\Rightarrow b = 1$Substituting $b = 1$ in (ii): $a + 4(1) = -5 \Rightarrow a = -9$Thus,the point is $(-9, 1)$.The value of $5a + b = 5(-9) + 1 = -45 + 1 = -44$.

If the point of intersection of the tangents drawn at the points where the line $5x + y + 1 = 0$ cuts the circle $x^2 + y^2 - 2x - 6y - 8 = 0$ is $(a, b)$,then $5a + b =$

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