Question

Question 60
The locus of the middle point of the chord of the circle
x + y2 = 1 such that the segment of the chord on the
parabola y = r * subtends a right angle at the origin, is
a circle whose centre and radius respectively are:​

Answer 1

Given : The locus of middle point of the chord of the circle x² + y² = 1 such that the segment of the chord on the parabola, y = x² - x subtends a right angle at the origin.

To find : we know, if (h, k) is the midpoint of chord of circle x² + y² = 1, then equation of chord is given by, T = S₁

⇒hx + ky - 1 = h² + k² - 1

⇒hx + ky = h² + k²

⇒1 = (hx + ky)/(h² + k²) ..........(1)

equation of parabola, y = x² - x

⇒y(1) = x² - x(1)

⇒y[(hx + ky)/(h² + k²)] = x² - x[(hx + ky)/(h² + k²)]

⇒hxy + ky² = x²(h² + k²) - hx² + kxy

⇒x² (h² + k² - h) + (-k)y² + xy(k - h) = 0

as parabola, y = x² - x subtends a right angle at the origin.

so, coefficient of x² + coefficient of y² = 0

⇒h² + k² - h - k = 0

putting, h = x , k = y

⇒x² + y² - x - y = 0

⇒x² - x + 1/4 + y² - y + 1/4 = 1/2

⇒(x - 1/2)² + (y - 1/2)² = (1/√2)²

it is clear that centre of circle is (1/2, 1/2) and radius of circle is 1/√2