QJEEM19Macb1b: JEE Main 2019 Mathematics - Class 11 JEEM19Macb1b Solution: A circle cuts a chord of length 4a on the x-axis and passes…

Let equation of circle be

x² + y² + 2gx + 2fy + c = 0

It passes through (0, 2b):

$$\Rightarrow$$  0 + (2b)² + 2g(0) + 2f(2b) + c = 0

$$\Rightarrow$$  4b² + 4bf + c = 0       . . . (i)

It cuts a chord of length 4a on x-axis:

$$2\sqrt{g^2-c}=4a$$

$$\Rightarrow$$  $$\sqrt{g^2-c}=2a$$

$$\Rightarrow$$  g² $$-$$ c = 4a² $$\Rightarrow$$ c = g² $$-$$ 4a²      . . . (ii)

Putting value of c in equation (i)

$$\Rightarrow$$  4b² + 4bf + g² $$-$$ 4a² = 0

For locus of centre (x, y), we know centre is ($$-$$g, $$-$$f). So, let x = $$-$$g and y = $$-$$f:

$$\Rightarrow$$  g = $$-$$x, f = $$-$$y

Substitute these in the equation:

$$\Rightarrow$$  4b² + 4b($$-$$y) + ($$-$$x)² $$-$$ 4a² = 0

$$\Rightarrow$$  x² $$-$$ 4by + 4(b² $$-$$ a²) = 0

It represents a Parabola.