JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Circle

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is

9 0^{\circ}

, then the length (in cm ) of their common chord is :

(A) $\frac{13}{2}$
(B) $\frac{120}{13}$ ✓ Correct
(C) $\frac{13}{5}$
(D) $\frac{60}{13}$

Answer: (B)

Solution

When two circles cut at right angles, the tangents at a crossing point are perpendicular, so the radii to that point and the line of centres form a right triangle: the distance between centres is

d = r_{1}^{2} + r_{2}^{2} = 5^{2} + 1 2^{2} = 13 cm

. Let the common chord have half-length

x

, at perpendicular distances

d_{1}

and

d_{2}

from the two centres with

d_{1} + d_{2} = 13

. From

d_{1}^{2} + x^{2} = 25

and

d_{2}^{2} + x^{2} = 144

, subtracting gives

d_{2}^{2} - d_{1}^{2} = 119

, and with

d_{2} = 13 - d_{1}

this yields

d_{1} = \frac{25}{13}

. Then

x^{2} = 25 - (\frac{25}{13})^{2} = \frac{25 \cdot 144}{169}

, so

x = \frac{60}{13}

and the full chord is

2 x = \frac{120}{13} cm

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Single correct option

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JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Circle

Solution

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