Question

[Key Concept: A chord of length $a$ on a line with $y$-intercept $c$: use the chord-length formula $2\sqrt{r^{2} - p^{2}} = a$.]

Given circle:

x² + y² = a²

Let the equation of the line be: y = mx + c or mx − y + c = 0

Here, c is the y-intercept.

The distance of the centre O(0, 0) from the line is:

d = |c|/√(m² + 1)

For a circle of radius a, if a chord has length L, then:

L = 2√(a² − d²)

Given:

L = a

Therefore,

a = 2√(a² − d²)

Squaring both sides:

a² = 4(a² − d²)

a² = 4a² − 4d²

4d² = 3a²

Therefore, d² = 3a²/4

But, d² = c²/(m² + 1)

Hence, c²/(m² + 1) = 3a²/4

Therefore, c² = 3a²(m² + 1)/4

Since m² ≥ 0,

m² + 1 ≥ 1

Therefore, c² ≥ 3a²/4

or, 4c² ≥ 3a²

Hence, the correct option is: