Graph of polynomials

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Graph of polynomials

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#1 h Apr 20, 2025, 3:00 PM

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The graph of the quadratic polynomial with real coefficients y = px^2 + qx + r, called G1, intersects the graph of the polynomial y = x^2, called G2, at points A and B. The lines tangent to G2 at points A and B intersect at point C. It is known that point C lies on G1. What is the value of p?

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#2 h Apr 20, 2025, 5:36 PM

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Given the parabola $G_{2}:y=x^{2}$ and the points $A(\lambda,\lambda^{2}),B(\mu,\mu^{2})$ .
The tangent lines $y=2\lambda x-\lambda^{2}$ and $y=2\mu x-\mu^{2}$ intersect in the point $C(\frac{\lambda+\mu}{2},\lambda \mu)$ .

The three points $A,B,C$ lie on $G_{1}:y=px^{2}+qx+r$ , giving a system of three equations with three unknowns.
Solving: $p=2,q=-\lambda-\mu,r=\lambda \mu$ .

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