Question

The number of points with integral coordinates that lie in the interior of the region common to the circle x² + y² = 16 and the parabola y² = 4x is –

• A. 8
• B. 10
• C. 16
• D. None of these

Answer 1

Answer: (A)

8

Answer 2

(l, µ) is interior to both the curves if

l² + µ² – 16 < 0 and µ² – 4l < 0.

Now, µ² – 4l < 0, Ž l > $\left( \frac{µ}{2} \right)^{2}$

Hence, if µ = 0, l = 1, 2, 3, ...; if µ = 1, l = 1, 2, 3,…;

if µ = 2, l = 2, 3, ...; if µ = 3, l = 3, 4,…

Also l² + µ² – 16 < 0 Ž l² < 16 – µ² .

Hence, if µ = 0, l = 1, 2, 3; if µ = 1, l = 1, 2, 3; if µ = 2,

l = 2, 3 ; if µ = 3, l has no integral value.

\ (1, 0), (2, 0), (3, 0), (1, 1), (2, 1), (3, 1), (2, 2), (3, 2) are the possible points.