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  • A circle with centre and radius 4 intersects the line <img alt="x+y=3" src="https://entrancecorner.oncodecogs.c

A circle with centre (2,3) and radius 4 intersects the line x+y=3 at the points P and Q. If the tangents at P and Qintersect at the point S(\alpha, \beta), then 4 \alpha-7 \beta is equal to

Option: 1

11


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\begin{aligned} & (x-2)^2+(y-3)^2=16 \\ & x^2+y^2-4 x-6 y-3=0 \end{aligned}

let the chord of contact w.r. to the point S(\alpha, \beta) is
$$ \begin{aligned} & T=0 \\ & \alpha x+\beta y-2(x+\alpha)-3(y+\beta)-3=0 \\ & x(\alpha-2)+y(\beta-3)-2 \alpha-3 \beta-3=0 \end{aligned}
Comparison with \mathrm{x}+\mathrm{y}=3

\frac{\alpha-2}{1}=\frac{\beta-3}{1}=\frac{-2 \alpha-3 \beta-3}{-3}

\begin{array}{ll} \alpha-2=\beta-3 & -3 \beta+9=-2 \alpha-3 \beta-3 \\ \alpha-\beta=-1 & 2 \alpha=-3-9 \\ \beta=\alpha+1 & \alpha=\frac{-12}{2} \\ =-6+1=-5 & \alpha=-6 \\ 4 \alpha-7 \beta & \\ 4(-6)-7(-5) & \\ \Rightarrow-24+35=11 & \end{array}

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