If $a^{2} + 14 a b + b^{2} = 12 h^{2}$ , then the angle between the lines represented by $a x^{2} + 2 h x y + b y^{2} = 0$ is

30^{o}

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60^{o}

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45^{o}

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90^{o}

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Solution

The correct option is D $30^{o}$
If angle between the two lines is $θ$ , then
$tan θ = ∣ ∣ ∣ \frac{2 \sqrt{h^{2} - a b}}{a + b} ∣ ∣ ∣$

It is given that,
$12 h^{2} = a^{2} + b^{2} + 14 a b$

$h^{2} = \frac{a^{2} + b^{2} + 14 a b}{12}$

So,
$tan θ = ∣ ∣ ∣ ∣ ∣ ∣ ∣ \frac{2 \sqrt{\frac{a^{2} + b^{2} + 14 a b}{12} - a b}}{a + b} ∣ ∣ ∣ ∣ ∣ ∣ ∣$

$= ∣ ∣ ∣ \frac{2 \sqrt{a^{2} + b^{2} + 2 a b}}{\sqrt{12} (a + b)} ∣ ∣ ∣$

$= ∣ ∣ ∣ ∣ \frac{\sqrt{(a + b)^{2}}}{\sqrt{3} (a + b)} ∣ ∣ ∣ ∣$

$= \frac{1}{\sqrt{3}}$

$\Rightarrow θ = 30^{\circ}$

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