If the equation of the normal to the curve $y = \frac { x - a } { ( x + b ) ( x - 2 ) }$ at the point $( 1 , - 3 )$ is $x - 4 y = 13$ then the value of $a + b$ is equal to $\_\_\_\_$

Let $f ( x ) = x ^ { 4 } + 2 x ^ { 3 } - 12 x ^ { 2 } + 4$. We are to find the values of $p$ such that we can draw three tangents to the curve $y = f ( x )$ from the point $\mathrm { P } ( 0 , p )$ on the $y$-axis.
(i) The equation of the tangent to the curve $y = f ( x )$ at the point $( t , f ( t ) )$ is
$$y = \left( \mathbf { A } t ^ { 3 } + \mathbf { B } t ^ { 2 } - \mathbf { C D } t \right) x - \mathbf { E } t ^ { 4 } - \mathbf { F } t ^ { 3 } + \mathbf { G H } t ^ { 2 } + \mathbf { I }$$
The condition under which this straight line passes through the point $\mathrm { P } ( 0 , p )$ is that
$$p = - \mathbf { J } t ^ { 4 } - \mathbf { K } t ^ { 3 } + \mathbf { L M } t ^ { 2 } + \mathbf { N }$$
(ii) For $\mathbf { O }$ and $\mathbf { S }$ in the following statements, choose either (0) or (1) and for the other blanks, enter the correct number. (0) local minimum
(1) local maximum
When the right side of (1) is set to $g ( t )$, the function $g ( t )$ takes a $\mathbf{O}$ at $t = \mathbf { P Q }$ and $t = \mathbf { R }$. On the other hand, $g ( t )$ takes a $\mathbf { S }$ at $t = \mathbf { T }$.
Hence the values of $p$ such that we can draw three tangents to the curve $y = f ( x )$ from the point $\mathrm { P } ( 0 , p )$ are
$$p = \mathbf { U } \text { and } p = \mathbf { V } ,$$
where $\mathbf { U } < \mathbf { V }$.

Consider the function $f(x) = x^3 - 4x + 4$. Let the straight line $\ell$ be the tangent to the graph of $y = f(x)$ at the point $\mathrm{A}(-1, 7)$, and the straight line $m$ be the tangent to the graph of $y = f(x)$ that passes through the point $\mathrm{B}(0, -12)$. Also, let C be the point of intersection of $\ell$ and $m$. Let us denote the angle formed by $\ell$ and $m$ at C by $\theta$ $\left(0 < \theta < \frac{\pi}{2}\right)$. We are to find $\tan\theta$.
(1) The derivative $f'(x)$ of $f(x)$ is $$f'(x) = \mathbf{A}x^{\mathbf{B}} - \mathbf{C}.$$ Hence, the slope of $\ell$ is $\mathbf{DE}$, and the equation of $\ell$ is $$y = \mathbf{DE}x + \mathbf{F}.$$
(2) Let us denote by $a$ the $x$-coordinate of the tangent point of the graph of $y = f(x)$ and line $m$. Then the equation of $m$ can be expressed in terms of $a$ as $$y = (\mathbf{G}a^{\mathbf{H}} - \mathbf{I})x - \mathbf{J}a^{\mathbf{K}} + \mathbf{K}.$$ Since line $m$ passes through point $\mathrm{B}(0, -12)$, we see that $a = \mathbf{M}$, and the equation of $m$ is $$y = \mathbf{N}x - \mathbf{OP}.$$ Hence, the coordinates of point C, the intersection of $\ell$ and $m$, are $(\mathbf{Q}, \mathbf{R})$.
(3) Let us denote by $\alpha$ the angle between the positive direction of the $x$-axis and line $\ell$, and by $\beta$ the angle between the positive direction of the $x$-axis and line $m$. Then we see that $$\tan\alpha = \mathbf{ST}, \quad \tan\beta = \mathbf{U},$$ and hence $$\tan\theta = \frac{\mathbf{V}}{\mathbf{W}}.$$

Let $a , b$ be real numbers, and let $O$ be the origin of the coordinate plane. It is known that the graph of the quadratic function $f ( x ) = a x ^ { 2 }$ and the circle $\Omega : x ^ { 2 } + y ^ { 2 } - 3 y + b = 0$ both pass through point $P \left( 1 , \frac { 1 } { 2 } \right)$, and let point $C$ be the center of $\Omega$.
Prove that the graph of $y = f ( x )$ and $\Omega$ have a common tangent line at point $P$.

On the coordinate plane, let $\Gamma$ be the graph of the cubic function $f(x) = x^{3} - 9x^{2} + 15x - 4$. Show that $P(1, 3)$ is a point on $\Gamma$, and find the equation of the tangent line $L$ to $\Gamma$ at point $P$.

For the function $f(x) = 2x^{3} - ax^{2} + 3$, what should $a$ be so that the equation of the tangent line to the curve at some point is $y = 4$?
A) $-3$
B) $-1$
C) $0$
D) $1$
E) $3$

The tangent line drawn from a point $A(x, y)$ on the parabola $y^{2} = 4x$ has slope 1.
Accordingly, what is $x + y$, the sum of the coordinates of point $A$?
A) 1
B) 2
C) 3
D) 4
E) 5

If the line tangent to the parabola $y = x^{2} + bx + c$ at the point $x = 2$ is $y = x$, what is the sum $b + c$?
A) $-2$
B) $-1$
C) $0$
D) $1$
E) $2$

At what point does the tangent line to the curve $\mathbf { y } = \sin ( \pi \mathrm { x } ) + \mathrm { e } ^ { \mathrm { x } }$ at $\mathrm { x } = 1$ intersect the y-axis?
A) $- \pi$
B) - 1
C) 0
D) $e - 1$
E) $\pi$

Given that the function $f$ has derivative $f ^ { \prime } ( x ) = 3 x ^ { 2 }$ and the tangent line at the point $x = a ( a > 0 )$ is the line $y - 12 x + 14 = 0$, what is the value of $f ( 1 )$?
A) $- 2$
B) 0
C) 1
D) 3
E) 5

A third-degree polynomial $\mathbf { P } ( \mathbf { x } )$ with leading coefficient 1 and its derivative $P ^ { \prime } ( x )$ satisfy
$$P ( 0 ) = P ( 1 ) = P ^ { \prime } ( 1 ) = 0$$
Accordingly, what is the value of $P ( - 1 )$?
A) 3
B) 1
C) 0
D) - 2
E) - 4

The tangent line drawn to the graph of the function $y = f ( x )$ at the point $( 2,4 )$ passes through the point $( - 1,3 )$.
Accordingly, what is the value of $f ^ { \prime } ( 2 )$?
A) $\frac { 1 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 3 } { 5 }$

$$x ^ { 2 } - y ^ { 2 } = 1$$
What is the distance between the points where the lines tangent to the hyperbola curve and having slope 3 intersect the y-axis, in units?
A) $\sqrt { 2 }$
B) $2 \sqrt { 2 }$
C) $4 \sqrt { 2 }$
D) $\sqrt { 3 }$
E) $2 \sqrt { 3 }$

The line $y = 4 x - 2$ is tangent to the graph of the function $f ( x ) = x ^ { 4 } + 1$ at the point $\mathrm { P } ( \mathrm { a } , \mathrm { b } )$.
Accordingly, what is the sum $a + b$?
A) 3
B) 4
C) 5
D) 6
E) 7

Let a and b be real numbers. In the rectangular coordinate plane, the parabola
$$y = a x ^ { 2 } + b x$$
passes through the point $( 1,2 )$, and the tangent line to the parabola at this point intersects the y-axis at the point $( 0,1 )$.
Accordingly, what is the product $a \cdot b$?
A) - 3
B) - 2
C) - 1
D) 2
E) 4

Let a and b be real numbers, and $$f ( x ) = a \cdot \ln x + b \cdot x ^ { 2 } + 3$$ The equation of the tangent line drawn to the graph of the function at the point $(1, f(1))$ is given as $y - 2x + 1 = 0$.
Accordingly, what is the product $\mathbf{a} \cdot \mathbf{b}$?\ A) $- 18$\ B) $- 16$\ C) $- 12$\ D) $- 8$\ E) $- 6$

Let a, b and c be real numbers. The equation of the tangent line to the curve
$$y = \frac { a } { x + a }$$
at point $P ( a , b )$ is given in the form
$$y = \frac { - x } { 8 } + c$$
Accordingly, what is the sum $a + b + c$?
A) $\frac { 7 } { 4 }$ B) $\frac { 11 } { 4 }$ C) $\frac { 13 } { 4 }$ D) 2 E) 3

In the rectangular coordinate plane, the tangent line drawn to the graph of the function $f ( x ) = x ^ { 2 } + a x$ at the point $( 2 , f ( 2 ) )$ is tangent to the graph of the function $g ( x ) = b x ^ { 3 }$ at the point $( 1 , g ( 1 ) )$. Accordingly, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) 2
B) 4
C) 6
D) 8
E) 10