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Chapter 11: Q. 94 (page 775)

In Problems 92鈥�98, use Descartes鈥檚 method from Problem 91 to find the equation of the line tangent to each graph at the given point.

94.x2+y=5; at(-2,1).

Short Answer

Expert verified

The equation of the tangent line is4x-y+9=0.

Step by step solution

01

Step 1. Given information

The general form of the equation of the tangent line must be y=mx+b.

Substitute (-2,1)in y=mx+b.

1=m(-2)+b1+2m=bb=2m+1

Substitute b=2m+1in y=mx+b.

y=mx+(2m+1)

So the system of equations here is

y=mx+(2m+1)x2+y=5.

02

Step 2.  Solve the system of equations.

Substitute y=mx+(2m+1)in x2+y=5.

role="math" localid="1646976038805" x2+mx+(2m+1)=5x2+mx+(2m-4)=0

Using the formula for quadratic equation,

role="math" x=-m卤m2-4(1)(2m-4)2(1)=-m卤m2-4(2m-4)2

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

m2-4(2m-4)=0m2-8m+16=0(m-4)2=0 m-4=0m=4

03

Step 3. Substitute m=4 in y=mx+b.

We get

y=4x+(2脳4+1)y=4x+94x-y+9=0

So the equation of the tangent line forx2+y=5at(-2,1)is4x-y+9=0.

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Most popular questions from this chapter

The product of two numbers is 10, and the difference of their squares is 21. Find the numbers.

Solve each system of equations. If the system has no solution, say that it is inconsistent.

2x-y=04x+2y=12

Solve each system of equations. If the system has no solution, say that it is inconsistent.
x+3y=52x-3y=-8

Graph each inequality by hand.

3x+2y鈮�6

Verify that the values of the variables listed are solutions of the system of equations.

x-y=312x+y=3x=4,y=1;4,1
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