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Maths-

General

Easy

Question

Let $P left parenthesis 2 comma blank 2 right parenthesis$ and $Q left parenthesis 1 divided by 2 comma blank minus 1 right parenthesis$ be two points on the parabola $y to the power of 2 end exponent equals 2 x$ . The coordinates of the point $R$ on the parabola $y to the power of 2 end exponent equals 2 x$ , where the tangent to the curve is parallel to the chord $P Q$ , are

$(2, - 1)$
$(\frac{1}{8}, \frac{1}{2})$
$\sqrt{2}, 1)$
$- \sqrt{2}, 1)$

The correct answer is: $open parentheses fraction numerator 1 over denominator 8 end fraction comma fraction numerator 1 over denominator 2 end fraction close parentheses$

Let $R left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis$ be the point on the parabola $y to the power of 2 end exponent equals 2 x$ such that tangent at $R$ is parallel to the chord $P Q$
$therefore open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript R end subscript equals S l o p e blank o f blank P Q$

$rightwards double arrow fraction numerator 1 over denominator y subscript 1 end subscript end fraction equals fraction numerator negative 1 minus 2 over denominator fraction numerator 1 over denominator 2 end fraction minus 2 end fraction open square brackets because y to the power of 2 end exponent equals 2 x rightwards double arrow 2 y fraction numerator d y over denominator d x end fraction equals 2 rightwards double arrow fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator y end fraction close square brackets$

$rightwards double arrow fraction numerator 1 over denominator y subscript 1 end subscript end fraction equals 2 rightwards double arrow y subscript 1 end subscript equals fraction numerator 1 over denominator 2 end fraction$

Since $open parentheses x subscript 1 end subscript comma blank y subscript 1 end subscript close parentheses$ lies on $y to the power of 2 end exponent equals 2 x$

$therefore y subscript 1 end subscript superscript 2 end superscript equals 2 x subscript 1 end subscript rightwards double arrow x subscript 1 end subscript equals fraction numerator 1 over denominator 8 end fraction open square brackets because y equals 1 divided by 2 close square brackets$

Hence, the required point is (1/8, 1/2)

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