Q68 / 200 · TS EAMCET 2001 · Mathematics

If one of the lines of pair of straight lines $a x^2+2 h x y+b y^2=0$ bisects the angle between the coordinate axes, then

  • A
    $a^2+b^2=h^2$
  • Correct answer
    B
    $(a+b)^2=4 h^2$
  • C
    $a^2+b^2=4 h^2$
  • D
    $(a+b)^2=h^2$

Solution

The given pair of straight lines is $\begin{array}{rlrl} & a x^2+2 h x y+b y^2 & =0 \\ \therefore & & m_1+m_2 & =\frac{-2 h}{b} \\ & \text { and } & m_1 m_2 & =\frac{a}{b}\end{array}$ $m_1=\tan 45^{\circ}=1$ Here, Then, the equation of the line is $y=x$. We have, $\begin{array}{rlrl} & & a x^2+2 h x^2+b x^2 & =0 \\ \Rightarrow & a+2 h+b & =0 \\ \Rightarrow & a+b & =-2 h \\ \Rightarrow & & a+b)^2 & =4 h^2\end{array}$

Hence, the correct answer is option: 'B'.

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