If the curve y = ax² + bx + c, x∈R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are :

Solution

Maths sir · 4 steps

A tangent at the origin tells you two things: the curve passes through (0,0) and its slope there is 1.

1. Through the origin. The tangent touches the curve at the origin, so the curve itself passes through (0, 0). Putting x = 0, y = 0 in y = ax² + bx + c gives c = 0.
2. Slope at origin. dy/dx = 2ax + b, so the slope at x = 0 is b. The tangent y = x has slope 1, so b = 1.
3. Use point (1, 2). Putting x = 1, y = 2 gives a + b + c = 2. With b = 1 and c = 0, this gives a = 1.
4. So. a = 1, b = 1, c = 0, so the curve is y = x² + x. That is option (b).

The idea to keep. The tangent at a point gives you two equations: the point lies on the curve, and dy/dx there equals the tangent's slope. Use both before you use the extra point.

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