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Maths-

General

Easy

Question

The two tangents to the curve $a x to the power of 2 end exponent plus 2 h x y plus b y to the power of 2 end exponent equals 1 comma blank a greater than 0$ at the points where it crosses $x$ -axis, are

Parallel
Perpendicular
Inclined at an angle of $π / 4$
None of these

The correct answer is: Parallel

At the point where the given curve crosses $x$ -axis, we have
$y equals 0$

$rightwards double arrow a x to the power of 2 end exponent equals 1$ [Putting $y equals 0$ in $a x to the power of 2 end exponent plus 2 h x y plus b y to the power of 2 end exponent equals 1$ ]

$rightwards double arrow x equals plus-or-minus fraction numerator 1 over denominator square root of a end fraction$

Thus, the given curve cuts $x$ -axis at $P left square bracket left parenthesis 1 divided by square root of a comma blank 0 right parenthesis$ and $Q open parentheses negative 1 divided by square root of a comma blank 0 close parentheses$

Now,

$a x to the power of 2 end exponent plus 2 h x y plus b y to the power of 2 end exponent equals 1$

$rightwards double arrow 2 a x plus 2 h open parentheses x fraction numerator d y over denominator d x end fraction plus y close parentheses plus 2 b y fraction numerator d y over denominator d x end fraction equals 0$

$rightwards double arrow fraction numerator d y over denominator d x end fraction equals negative fraction numerator a x plus h y over denominator h x plus b y end fraction$

$rightwards double arrow open parentheses fraction numerator d y over denominator x end fraction close parentheses subscript P end subscript equals negative fraction numerator a over denominator h end fraction blank a n d open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript Q end subscript equals negative fraction numerator a over denominator h end fraction rightwards double arrow open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript P end subscript equals open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript Q end subscript$

Hence, the tangents are parallel

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