A line passing through the point of intersection of \(x+y=4\) and \(x-y=2\) makes an angle \(\tan ^{-1}\left(\frac{y}{4}\right)\) with the \(x\)-axis. It intersects the parabola \(y^{2}=4(x-3)\) at points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) respectively. Then \(\left|x_{1}-x_{2}\right|\) is equal to

Question

A line passing through the point of intersection of \(x+y=4\) and \(x-y=2\) makes an angle tan \(^{-1}\left(\frac{3}{4}\right)\) with the \(x\)-axis. It intersects the parabola \(y^{2}=4(x-3)\) at points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) respectively. Then \(\left|x_{1}-x_{2}\right|\) is equal to
(A) \(\frac{16}{9}\)
(B) \(\frac{32}{9}\)
(C) \(\frac{40}{9}\)
(D) \(\frac{80}{9}\)

Answer 1

Ans: (B)
 Point of intersection of \(x+y=4\) and \(x-y=2\) is \(\equiv(3,1)\)
The line though this making an angle \(\tan ^{-1} \frac{3}{4}\) with the \(x\)-axis
is \((y-1)=\frac{3}{4}(x-3)\)
\(\Rightarrow y=\frac{3 x}{4}-\frac{5}{4}=\frac{3 x-5}{4}\)
Putting \(y\) in \(y^{2}=4(x-3)\), we have
\(9 x^{2}-94 x+217=0\)
\(\Rightarrow x_{1}+x_{2}=\frac{94}{9} \quad\) and \(x_{1} x_{2}=\frac{217}{9}\) \(\Rightarrow\left|x_{1}-x_{2}\right|=\sqrt{\left(x_{1}+x_{2}\right)^{2}-4 x_{1} x_{2}}=\frac{32}{9}\)

Answer 2

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