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The pair of straight lines represented by the equation \(3 d x^2-5 x y+\left(d^2-2\right) y^2=0\). If the lines are perpendicular to each other, for how many values of \(d\) this condition will be satisfied?
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Verified Answer
The correct answer is:
2
Given, pair of straight lines is
\(3 d x^2-5 x y+\left(d^2-2\right) y^2=0\)
Since, lines are perpendicular
\(\Rightarrow x^2\) coefficient \(+y^2\) coefficient \(=0\)
\(\begin{gathered}
3 d+d^2-2=0 \\
d^2+3 d-2=0 \\
d=\frac{-3 \pm \sqrt{9-4 \cdot(-2)}}{2 \cdot 1}=\frac{-3 \pm \sqrt{17}}{2} \\
=\frac{-3+\sqrt{17}}{2}(\text { or })=\frac{-3-\sqrt{17}}{2}
\end{gathered}\)
Number of possible value of \(d\) are 2 Hence, option (b) is correct.
\(3 d x^2-5 x y+\left(d^2-2\right) y^2=0\)
Since, lines are perpendicular
\(\Rightarrow x^2\) coefficient \(+y^2\) coefficient \(=0\)
\(\begin{gathered}
3 d+d^2-2=0 \\
d^2+3 d-2=0 \\
d=\frac{-3 \pm \sqrt{9-4 \cdot(-2)}}{2 \cdot 1}=\frac{-3 \pm \sqrt{17}}{2} \\
=\frac{-3+\sqrt{17}}{2}(\text { or })=\frac{-3-\sqrt{17}}{2}
\end{gathered}\)
Number of possible value of \(d\) are 2 Hence, option (b) is correct.
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