Chapter 11: Q. 93 (page 775)

In Problems 92–98, use Descartes’s method from Problem 91 to find the equation of the line tangent to each graph at the given point.
93. $y = x^{2} + 2$ ; at $(1, 3)$ .

Short Answer

Expert verified

The equation of the tangent line is $y = 2 x + 1$ .

Step by step solution

Step 1. Given information

The general form of the equation of the tangent must be $y = m x + b$ .

Substitute $(x, y) = (1, 3)$ in $y = m x + b$ .

$3 = m (1) + b 3 - m = b b = 3 - m$

Substitute $b = 3 - m$ in $y = m x + b$ .

$y = m x + (3 - m)$ .

So the system of equations here is

$y = m x + (3 - m) y = x^{2} + 2$ .

Step 2. Solve the system of equations obtained.

Substitute $y = m x + (3 - m)$ in $y = x^{2} + 2$ .

$m x + (3 - m) = x^{2} + 2 m x + 3 - m = x^{2} + 2 0 = x^{2} - m x + (m - 1) x^{2} - m x + (m - 1) = 0$

Using the quadratic formula,

role="math" localid="1646960621570" $x = \frac{m x \pm \sqrt{{(m)}^{2} - 4 (1) (m - 1)}}{2 (1)} x = \frac{m x \pm \sqrt{m^{2} - 4 (m - 1)}}{2}$

For the unique solution, the two roots must be equal which means

$m^{2} x^{2} - 4 (m - 1) = 0 m^{2} - 4 m + 4 = 0 {(m - 2)}^{2} = 0 m - 2 = 0 m = 0$

Step 3. Substitute m=2 in y=mx+b.

We get

$y = 2 x + (3 - 2) y = 2 x + 1$

So the equation of the tangent line of $y = x^{2} + 2$ at $(1, 3)$ is $y = 2 x + 1$ .

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In Problems 92–98, use Descartes’s method from Problem 91 to find the equation of the line tangent to each graph at the given point.
93. $y = x^{2} + 2$ ; at $(1, 3)$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve the system of equations obtained.

Step 3. Substitute m=2 in y=mx+b.

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In Problems 92–98, use Descartes’s method from Problem 91 to find the equation of the line tangent to each graph at the given point.93.y=x2+2; at(1,3).

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve the system of equations obtained.

Step 3. Substitute m=2 in y=mx+b.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Statistics

Discrete Mathematics

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

In Problems 92–98, use Descartes’s method from Problem 91 to find the equation of the line tangent to each graph at the given point.
93. $y = x^{2} + 2$ ; at $(1, 3)$ .