Applications of Derivatives
Tangent and Normal to Polynomial Curves
nta_pyq_2023_apr
Grade 12
Question:
Let the quadratic curve passing through the point $(-1,0)$ and touching the line $y=x$ at $(1,1)$ be $y=f(x)$. Then the $x$-intercept of the normal to the curve at the point $(\alpha,\alpha+1)$ in the first quadrant is _______.
Step-by-Step Solution
Key Concept: Let $f(x)=ax^2+bx+c$. Three conditions: passes through $(-1,0)$, passes through $(1,1)$, and $f'(1)=1$ (tangent slope equals slope of $y=x$). Solve for $a,b,c$.
$a=\frac{1}{4},b=\frac{1}{2},c=\frac{1}{4}$. Curve: $y=\frac{(x+1)^2}{4}$. For $(\alpha,\alpha+1)$: $\alpha+1=\frac{(\alpha+1)^2}{4}\Rightarrow\alpha=3$. $f'(3)=2$, normal slope $=-\frac{1}{2}$. Normal: $y-4=-\frac{1}{2}(x-3)\Rightarrow x$-intercept $=11$.
Correct Answer: 11