Suppose $d_1$ and $d_2$ are respectively the lengths of intercepts of the cirlce $x^2+y^2=4$ and $x^2+y^2-10 x-14 y+65=0$ on the line $2 x-2 y-3=0$. Then, which of the following is true?

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Aicharya · Accepted Answer

$\begin{aligned} & O A=\frac{|0-0-3|}{\sqrt{2^2+2^2}}=\frac{3}{\sqrt{8}} \\ & A B=\sqrt{2^2-\left(\frac{3}{\sqrt{8}}\right)^2}=\sqrt{4-\frac{9}{8}}=\sqrt{\frac{23}{8}}\end{aligned}$ $\therefore \quad d_1=2 \sqrt{\frac{23}{8}}$

$\Rightarrow x^2+y^2-10 x-14 y+65=0$ $\begin{gathered}g=-5, f=-7, c=65 \\ r=\sqrt{25+44-65}=3\end{gathered}$ $\therefore \quad P Q=\frac{|2 \times 5-2 \times 7-3|}{\sqrt{2^2+2^2}}=\frac{7}{\sqrt{8}}$ $Q R=\sqrt{3^2-\left(\frac{7}{\sqrt{2}}\right)^2}=\sqrt{\frac{23}{8}}$ $d_2=2 Q R \Rightarrow d_2=2 \sqrt{\frac{23}{8}}$ $\therefore \quad d_1=d_2$

Suppose $d_1$ and $d_2$ are respectively the lengths of intercepts of the cirlce $x^2+y^2=4$ and $x^2+y^2-10…

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