Key Concepts and Formulas

• Derivative as Slope of Tangent: The derivative of a function $y = f(x)$, denoted by $\frac{dy}{dx}$, gives the slope of the tangent line to the curve at the point $(x, y)$.
• Slope of a Line: The slope of a line given by $Ax + By + C = 0$ is $m = -\frac{A}{B}$. Alternatively, rewriting the equation in slope-intercept form $y = mx + c$ directly gives the slope $m$.
• Perpendicular Lines: If two lines with slopes $m_1$ and $m_2$ are perpendicular, then $m_1 m_2 = -1$.
• Point on a Curve: A point $(x_0, y_0)$ lies on the curve $y = f(x)$ if and only if $y_0 = f(x_0)$.

Step-by-Step Solution

Step 1: Find the derivative of the curve and evaluate it at x = 1

• We are given the curve $y = x^3 + ax - b$. We need to find the slope of the tangent at the point (1, -5). The slope of the tangent is given by the derivative $\frac{dy}{dx}$.
• Differentiate $y$ with respect to $x$: $\frac{dy}{dx} = \frac{d}{dx}(x^3 + ax - b) = 3x^2 + a$
• Evaluate the derivative at $x = 1$: $m_1 = \frac{dy}{dx}\Big|_{x=1} = 3(1)^2 + a = 3 + a$
• The derivative gives us the slope of the tangent at any point on the curve. Evaluating it at x=1 gives us the slope of the tangent at the specific point (1,-5).

Step 2: Find the slope of the given line

• We are given the line $-x + y + 4 = 0$. We need to find its slope.
• Rewrite the equation in slope-intercept form $y = mx + c$: $y = x - 4$
• The slope of the line is $m_2 = 1$.
• Alternatively, using the formula $m = -\frac{A}{B}$ for the line $Ax + By + C = 0$, we have $A = -1$ and $B = 1$, so $m_2 = -\frac{-1}{1} = 1$.
• Knowing the slope of this line is crucial since the tangent to the curve is perpendicular to it.

Step 3: Use the perpendicularity condition to find 'a'

• The tangent to the curve is perpendicular to the line $-x + y + 4 = 0$.
• The product of the slopes of perpendicular lines is -1, so $m_1 m_2 = -1$.
• We have $m_1 = 3 + a$ and $m_2 = 1$, so: $(3 + a)(1) = -1$ $3 + a = -1$ $a = -4$
• The perpendicularity condition allows us to find the value of the unknown constant 'a'.

Step 4: Use the point (1, -5) to find 'b'

• Now we know $a = -4$, so the equation of the curve is $y = x^3 - 4x - b$.
• The point (1, -5) lies on the curve, so we can substitute $x = 1$ and $y = -5$ into the equation: $-5 = (1)^3 - 4(1) - b$ $-5 = 1 - 4 - b$ $-5 = -3 - b$ $b = 2$
• Since the point (1,-5) lies on the curve, it must satisfy the equation of the curve. This allows us to solve for the unknown constant 'b'.

Step 5: Write the complete equation of the curve

• We have found $a = -4$ and $b = 2$, so the equation of the curve is: $y = x^3 - 4x - 2$

Step 6: Check which of the given options lies on the curve

• We need to check each option to see if it satisfies the equation $y = x^3 - 4x - 2$.

  • (A) (2, -2) Substitute $x = 2$: $y = (2)^3 - 4(2) - 2 = 8 - 8 - 2 = -2$ Since $y = -2$, the point (2, -2) lies on the curve.

  • (B) (2, -1) Substitute $x = 2$: $y = (2)^3 - 4(2) - 2 = 8 - 8 - 2 = -2$ Since $y = -2 \neq -1$, the point (2, -1) does not lie on the curve.

  • (C) (-2, 2) Substitute $x = -2$: $y = (-2)^3 - 4(-2) - 2 = -8 + 8 - 2 = -2$ Since $y = -2 \neq 2$, the point (-2, 2) does not lie on the curve.

  • (D) (-2, 1) Substitute $x = -2$: $y = (-2)^3 - 4(-2) - 2 = -8 + 8 - 2 = -2$ Since $y = -2 \neq 1$, the point (-2, 1) does not lie on the curve.

• Only the point (2, -2) satisfies the equation of the curve.

Common Mistakes & Tips

• Sign Errors: Be careful with signs when calculating slopes and substituting values.
• Perpendicularity Condition: Remember that the product of the slopes of perpendicular lines is -1.
• Point of Tangency: The point of tangency always lies on the curve.

Summary

We first found the derivative of the curve to determine the slope of the tangent at x=1. Then, we found the slope of the given line. Using the condition for perpendicular lines, we solved for 'a'. Knowing that the point (1, -5) lies on the curve, we substituted it into the equation and solved for 'b'. Finally, we tested each of the given points to see which one satisfies the equation of the curve $y = x^3 - 4x - 2$. The point (2, -2) is the only one that lies on the curve.

Final Answer

The final answer is $\boxed{(2, -2)}$, which corresponds to option (A).