Vectors + Tangents #1

Geometry Level 4

In the rectangular Cartesian system of coordinates $Oxy$ , a tangent is drawn to the curve $y=x^2+x+10$ at a point A $(x_0,y_0)$ , where $x_0=1$ . The tangent cuts the $Ox$ axis at a point B . Find the scalar product of the vectors $\vec{OA}$ and $\vec{OB}$ . If your answer is $A$ , then enter the value of $A-145$ .

The answer is -148.

1 solution

Chan Tin Ping
Nov 4, 2017

First, we get $A (1,12)$ . Now, to find scalar product of $\vec{OA}$ and $\vec{OB}$ , we need to find the coordinate of point $B$ .

$\dfrac{\text{d}}{\text{d}x}y=2x+1$ . Hence, the gradient of tangent of $y$ at point A is $2(1)+1=3$ . The equation of the tangent is $\frac{y-12}{x-1}=3$ . Point $B$ lies on the tangent and its y-coordinate is $0$ . Let $B(x_0,0)$ , we get $x_0=-3$ .

Hence, $\vec{OA} =\binom{1}{12}$ and $\vec{OB} =\binom{-3}{0}$ . The scalar product is $1\times -3 + 12\times 0 = -3$ .

Answer: $-3-145=-148$

Simply great solution!(+1)

Rishu Jaar - 3 years, 7 months ago

Vectors + Tangents #1

The answer is -148.

1 solution

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