JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Circle

If the length of the chord of the circle,

x^{2} + y^{2} = r^{2} (r > 0)

along the line

y - 2 x = 3

r

, the

r^{2}

is equal to:

(A)12
(B) $\frac{12}{5}$ ✓ Correct
(C) $\frac{9}{5}$
(D) $\frac{24}{5}$

Answer: (B)

Solution

The chord that the line

y - 2 x = 3

cuts from

x^{2} + y^{2} = r^{2}

has length

r

, the same as the radius. The perpendicular distance from the centre to the line is

\frac{∣ - 3∣}{1 ^{2} + 2 ^{2}} = \frac{3}{5}

. Half the chord is

\frac{r}{2}

, and the perpendicular, the half-chord and the radius form a right triangle:

(\frac{3}{5})^{2} + (\frac{r}{2})^{2} = r^{2}

. Thus

\frac{9}{5} = \frac{3 r ^{2}}{4}

, giving

r^{2} = \frac{12}{5}

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Single correct option

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