Question

The equation \[2{x^2} + 4xy - p{y^2} + 4x + qy + 1 = 0\] will represent two mutually perpendicular straight lines. Which of the following is correct?
A. \[p = 1\] and \[q = 2\,\,{\rm{or}}\,\,6\]
B. \[p = 2\] and \[q = 0\,\,{\rm{or}}\,\,6\]
C. \[p = 2\] and \[q = 0\,\,{\rm{or}}\,\,8\]
D. \[p = - 2\] and \[q = - 2\,\,{\rm{or}}\,\,8\]

Answer 1

Hint: First we will compare the given pair of lines with \[a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0\]. Since the equation of lines is perpendicular lines then \[a + b = 0\]. By applying the formula \[a + b = 0\], we will calculate the value of \[p\]. Again apply the condition of the pair of lines \[abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0\] to calculate the value of \[q\].

Formula used:
The standard form of a pair of lines is \[a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0\].
If the pair of lines are perpendicular lines, then \[a + b = 0\].
The condition of the pair of lines is \[abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0\].

Complete step by step solution:
Given equation is \[2{x^2} + 4xy - p{y^2} + 4x + qy + 1 = 0\].
Compare the equation with \[a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0\]
\[a = 2\]
\[2h = 4 \Rightarrow h = 2\]
\[b = - p\]
\[2g = 4 \Rightarrow g = 2\]
\[2f = q \Rightarrow f = \dfrac{q}{2}\]
\[c = 1\]
Apply the condition of mutually perpendicular lines \[a + b = 0\]
\[2 - p = 0\] since \[a = 2\] and \[b = - p\]
\[ \Rightarrow p = 2\]
Apply the condition of pair of lines \[abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0\]
\[2 \cdot \left( { - p} \right) \cdot 1 + 2 \cdot \dfrac{q}{2} \cdot 2 \cdot 2 - 2{\left( {\dfrac{q}{2}} \right)^2} - \left( { - p} \right){\left( 2 \right)^2} - 1 \cdot {2^2} = 0\]
\[ \Rightarrow - 2p + 4q - \dfrac{{{q^2}}}{2} + 4p - 4 = 0\]
Now putting \[p = 2\]
\[ \Rightarrow - 2 \cdot 2 + 4q - \dfrac{{{q^2}}}{2} + 4 \cdot 2 - 4 = 0\]
\[ \Rightarrow - 4 + 4q - \dfrac{{{q^2}}}{2} + 8 - 4 = 0\]
\[ \Rightarrow 4q - \dfrac{{{q^2}}}{2} = 0\]
\[ \Rightarrow 8q - {q^2} = 0\]
Take common \[q\]
\[ \Rightarrow q\left( {8 - q} \right) = 0\]
Equate each factor with zero
\[q = 0\] \[8 - q = 0\]
\[ \Rightarrow q = 8\]
Thus \[p = 2\] and \[q = 0\] or 8.
Hence option C is the correct.

Note: Students often confused with equation of pair of lines \[a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0\] and \[a{x^2} + 2hxy + b{y^2} = 0\]. If the lines pass through the origin, then the equation of the pair of lines is \[a{x^2} + 2hxy + b{y^2} = 0\]. If the lines do not pass through the origin, then the equation of the pair of lines is \[a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0\].