If the line through the point P(5,3) meets th | vedclass

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(D) The point $A(4,2)$ lies on the circle $x^2+y^2-2x-4y+\alpha=0$. Substituting the coordinates of $A$ into the equation: $16+4-8-8+\alpha=0 \Rightar

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$8$

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(D) The point $A(4,2)$ lies on the circle $x^2+y^2-2x-4y+\alpha=0$. Substituting the coordinates of $A$ into the equation: $16+4-8-8+\alpha=0 \Rightarrow \alpha=-4$. The equation of the circle is $x^2+y^2-2x-4y-4=0$. The power of point $P(5,3)$ with respect to the circle is given by $PA \cdot PB$. The power of a point $(x_0, y_0)$ with respect to the circle $x^2+y^2+2gx+2fy+c=0$ is $x_0^2+y_0^2+2gx_0+2fy_0+c$. Substituting $P(5,3)$ into the circle equation: $PA \cdot PB = 5^2+3^2-2(5)-4(3)-4 = 25+9-10-12-4 = 8$.

If the line through the point $P(5,3)$ meets the circle $x^2+y^2-2x-4y+\alpha=0$ at $A(4,2)$ and $B(x_1, y_1)$,then $PA \cdot PB$ is equal to

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