If the line segment joining the points (5,2) and (2,a) subtends an angle π/4 at the origin, then the absolute value of the product of all possible values of a is :

Solution

Maths sir · 4 steps

Treat both points as vectors from the origin, use the dot product, and let Vieta give you the product.

1. Set up vectors. The angle at the origin is the angle between p⃗ = (5, 2) and q⃗ = (2, a). So cos π/4 = (p⃗ · q⃗)/(|p⃗| |q⃗|).

   (10 + 2a)/(√29 √(4 + a²)) = 1/√2

2. Square and simplify. Squaring gives 2(10 + 2a)² = 29(4 + a²), that is 200 + 80a + 8a² = 116 + 29a². This becomes 21a² - 80a - 84 = 0.
3. Check both roots. The roots are a = 14/3 and a = -6/7. Both make 10 + 2a > 0, so the dot product is positive and the angle is really π/4, not 3π/4. Both values are valid.
4. So. The product of the roots is -84/21 = -4, so its absolute value is 4, option (a).

The idea to keep. An angle subtended at the origin is just the angle between two position vectors. When you need a product of roots, form the quadratic and use Vieta. Check signs after squaring.

Answer from the NTA final answer key. Solution by PrepWiser.