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Chapter 21: Problem 52

The tangent to the curve \(y=x^{2}-5 x+5\), parallel to the line \(2 y=4 x+1\), also passes through the point : [Jan. 12, 2019 (II)] (a) \(\left(\frac{7}{2}, \frac{1}{4}\right)\) (b) \(\left(\frac{1}{8},-7\right)\) (c) \(\left(-\frac{1}{8}, 7\right)\) (d) \(\left(\frac{1}{4}, \frac{7}{2}\right)\)

Short Answer

Expert verified
(a) \(\left(\frac{7}{2}, \frac{1}{4}\right)\)

Step by step solution

01

Find the Slope of the Given Line

The equation of the given line is \(2y = 4x + 1\). Rewrite it in slope-intercept form \(y = mx + c\) to find the slope. Divide every term by 2: \(y = 2x + \frac{1}{2}\). Thus, the slope of this line is 2.
02

Find the Derivative of the Curve

The given curve is \(y = x^2 - 5x + 5\). Differentiate it with respect to \(x\) to find the slope of the tangent line: \(\frac{dy}{dx} = 2x - 5\). This gives the slope of the tangent at any point \((x, y)\) on the curve.
03

Equate the Slopes for Parallelism

For the tangent line to be parallel to the given line, set the derivative equal to the slope of the line: \(2x - 5 = 2\). Solve for \(x\): \(2x = 7\), hence \(x = \frac{7}{2}\).
04

Determine the Corresponding y-coordinate

To find the \(y\)-coordinate on the curve at \(x = \frac{7}{2}\), substitute \(x\) back into the curve equation: \(y = \left(\frac{7}{2}\right)^2 - 5\left(\frac{7}{2}\right) + 5\). Simplifying this gives \(y = \frac{49}{4} - \frac{35}{2} + 5\), which simplifies further to \(y = \frac{1}{4}\).
05

Identify the Point the Tangent Passes Through

The tangent at \(x = \frac{7}{2}\) on the curve is \(\left(\frac{7}{2}, \frac{1}{4}\right)\). Check which given option is \(\left(\frac{7}{2}, \frac{1}{4}\right)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola Equation
A parabola is a symmetric curve shaped like an arch. It's usually represented in the equation form \[ y = ax^2 + bx + c \]. This standard form highlights its quadratic nature, where:
  • a, b, and c are constants.
  • a dictates the direction and width of the parabola.
  • b and c adjust its position on the graph.
The given parabola equation from the problem is \( y = x^2 - 5x + 5 \). Parabolas have a vertex (highest or lowest point), and they open upwards if a is positive and downwards if it is negative. This particular parabola opens upwards because the coefficient of \( x^2 \), which is 1, is positive. Understanding this equation helps us find properties like the vertex, axis of symmetry, and focus.
Slope of a Line
The slope of a line is a measure of its steepness and direction. Often denoted by \( m \), it is calculated as the 'rise over run' or the change in y over the change in x. For a linear equation like \( y = mx + c \), m directly represents the slope. It tells us:
  • A positive \( m \) means the line ascends from left to right.
  • A negative \( m \) descends.
  • A larger absolute value of \( m \) indicates a steeper slope.
In the problem, the line original equation given is \( 2y = 4x + 1 \). Converting it into slope-intercept form gives \( y = 2x + \frac{1}{2} \), indicating a slope of \( 2 \). This tells us the line inclines upwards with moderate steepness.
Derivative of a Polynomial
Derivatives are a cornerstone of calculus that describe how a function changes. For polynomials, taking a derivative means applying the power rule: \[ \frac{d}{dx}(x^n) = nx^{n-1} \]. This rule simplifies calculating derivatives for polynomial terms. Given the curve \( y = x^2 - 5x + 5 \), we compute its derivative:
  • The derivative of \( x^2 \) is \( 2x \).
  • The derivative of \( -5x \) is \( -5 \).
  • Constants like \( 5 \) disappear when differentiated.
Thus, the derivative of the given function is \( \frac{dy}{dx} = 2x - 5 \). This derivative describes the slope of the tangent to the parabola at any point \( x \). Finding where derivatives equate to specific values helps in identifying points of interest, such as parallel tangents.
Parallel Lines
Parallel lines have the same slope but differ in their vertical position on a graph. This means that they keep the same distance between them and don't intersect. In the context of functions and curves:
  • For two lines to be parallel, they must have identical slopes.
  • In equations, if lines are in the form \( y = mx + c \), parallelism implies that \( m \) is the same.
When finding a tangent line to a curve that is parallel to another line, the slope of the tangent must equal the slope of the other line. In the exercise, two lines are parallel when their slopes are \( 2 \). Thus, solving \( 2x-5 = 2 \) helped us determine the point on the curve where the tangent line will parallel the given line, ensuring the correct positioning relative to specified points.

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Most popular questions from this chapter

The maximum area (in sq. units) of a rectangle having its base on the \(x\)-axis and its other two vertices on the parabola, \(\mathrm{y}=12-\mathrm{x}^{2}\) such that the rectangle lies inside the parabola, is: (a) 36 (b) \(20 \sqrt{2}\) (c) 32 (d) \(18 \sqrt{3}\)

If the tangent to the curve, \(y=x^{3}+a x-b\) at the point \((1,-5)\) is perpendicular to the line, \(-x+y+4=0\), then which one of the following points lies on the curve? [April 09, 2019 (I)] (a) \((-2,1)\) (b) \((-2,2)\) (c) \((2,-1)\) (d) \((2,-2)\)

Let \(f\) be any function continuous on \([a, b]\) and twice differentiable on \((a, b)\). If for all \(x \in(a, b), f 2(x)>0\) and \(f^{\prime \prime}(x)<0\), then for any \(c \in(a, b), \frac{f(c)-f(a)}{f(b)-f(c)}\) is greater than: \(\quad\) [Jan. 9, 2020 (I)] (a) \(\frac{b+a}{b-a}\) (b) 1 (c) \(\frac{b-c}{c-a}\) (d) \(\frac{c-a}{b-c}\)

If the volume of a spherical ball is increasing at the rate of \(4 \pi \mathrm{cc} / \mathrm{sec}\), then the rate of increase of its radius (in \(\mathrm{cm} / \mathrm{sec})\), when the volume is \(288 \pi \mathrm{cc}, \quad\) Online April 19, 2014] (a) \(\frac{1}{6}\) (b) \(\frac{1}{9}\) (c) \(\frac{1}{36}\) (d) \(\frac{1}{24}\)

If the surface area of a sphere of radius \(r\) is increasing uniformly at the rate \(8 \mathrm{~cm}^{2} / \mathrm{s}\), then the rate of change of its volume is: \(\quad\) [Online April 9, 2013] (a) constant (b) proportional to \(\sqrt{r}\) (c) proportional to \(r^{2}\) (d) proportional to \(r\)
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