The locus of the middle point of the chord of the circle $x^{2} + y^{2} = 1$ such that the segment of the chord on the parabola $y = x^{2} - x$ subtends a right angle at the origin, is a circle whose centre and radius respectively are

(1, 1) and

\sqrt{2}

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(1, 1) and 2

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(\frac{1}{2}, \frac{1}{2})

and

\frac{1}{2}

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(\frac{1}{2}, \frac{1}{2})

and

\frac{1}{\sqrt{2}}

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Solution

The correct option is C $(\frac{1}{2}, \frac{1}{2})$ and $\frac{1}{\sqrt{2}}$
Given : The locus of middle point of the chord of the circle $x^{2} + y^{2} = 1$

such that the segment of the chord on the parabola, $y = x^{2} - x$ subtends a right angle at the origin.

$⟹ h x + k y - 1 = h^{2} + k^{2} - 1 ⟹ h x + k y = h^{2} + k^{2} ⟹ 1 = \frac{(h x + k y)}{(h^{2} + k^{2})} - (1)$

Equation of parabola, $y = x^{2} - x$

$⟹ y (1) = x^{2} - x (1)$

$⟹ y [\frac{(h x + k y)}{(h^{2} + k^{2})}] = x^{2} - x [\frac{(h x + k y)}{(h^{2} + k^{2})}]$

$⟹ h x y + k y^{2} = x^{2} (h^{2} + k^{2}) - h x^{2} + k x y ⟹ x^{2} (h^{2} + k^{2} - h) + (- k) y^{2} + x y (k h) = 0$

As parabola, $y = x^{2} - x$ subtends a right angle at the origin.

so, coefficient of $x ²$ + coefficient of $y ² = 0$

$⟹ h^{2} + k^{2} - h - k = 0$

putting $h = x, k = y$

$⟹ x^{2} + y^{2} - x - y = 0 ⟹ x^{2} - x + \frac{1}{4} + y^{2} - y + \frac{1}{4} = \frac{1}{2}$

$⟹ (x - \frac{1}{2})^{2} + (y - \frac{1}{2})^{2} = (\frac{1}{\sqrt{2}}) ²$

It is clear that centre of circle is $(\frac{1}{2}, \frac{1}{2})$ and radius of circle is $\frac{1}{\sqrt{2}}$

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List- I	List- II
A. The locus of the point of intersection of the perpendicular tangents of $x^{2} + y^{2} = a^{2}$	1) $x^{2} + y^{2} = 5$ of
B. The locus of the mid-point of the chord of the circle $x^{2} + y^{2} = 8$ which is $\sqrt{2}$ units from origin is	2) $x^{2} + y^{2} = a^{2} {sec}^{2} α$
C. The locus of the point of intersection of the lines $x = \frac{1 - t^{2}}{1 + t^{2}}$ and $y = \frac{2 t}{1 + t^{2}}$	3) $x^{2} + y^{2} = 2 a^{2}$
D. The locus of the point whose chord of contact w.r.t $x^{2} + y^{2} = a^{2}$ making an angle '2 $α$ ' at the center is	4) $x^{2} + y^{2} = 2$ 5) $x^{2} + y^{2} = 1$

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