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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 96 (page 775)

Use Descartes鈥檚 method to find the equation of the line tangent to the graph $3 x^{2} + y^{2} = 7$ at the given point (-1,2).

Short Answer

Expert verified

Equation of tangent is $2 y = 3 x + 7$

Step by step solution

01

Step 1. Given information

$3 x^{2} + y^{2} = 7$ we have to find tangent at (-1,2)

02

Step 2. General equation of tangent

As we know that the equation of the tangent line is in the form of

y=mx +b

As this line is passes through (-1,2) So we get

$b = 2 + m$

Now we put b=2+m y=mx +b

We get

$y = m x + (2 + m)$

03

Step 3. Calculations

Now put $y = m x + (2 + m)$ in $3 x^{2} + y^{2} = 7$

We get

$3 x^{2} + [m x + (2 + m)]^{2} = 7 3 x^{2} + m^{2} x^{2} + 2 (m x) (2 + m) + (2 + m)^{2} = 7 x^{2} (3 + m^{2}) + x (4 m + 2 m^{2}) + (m^{2} + 4 m - 3) = 0$

04

Step 3. Solving the quadratic equatiom

Now we have to solve the quadratic equation . we get

$x = \frac{- (4 m + 2 m^{2}) 卤 \sqrt{{(4 m + 2 m^{2})}^{2} - 4 (3 + m^{2}) (m^{2} + 4 m - 3)}}{2 (3 + m^{2})}$

Now we want the unique solutions for this both the roots should be equal .So

${(4 m + 2 m^{2})}^{2} - 4 (3 + m^{2}) (m^{2} + 4 m - 3) = 0 16 m^{2} + 16 m^{3} + 4 m^{4} - 12 m^{2} - 48 m + 36 - 4 m^{4} - 16 m^{3} + 12 m^{2} = 0 16 m^{2} - 48 m + 36 = 0 (2 m - 3)^{2} = 0 m = \frac{3}{2}$

05

Step 5. Equation of tangent

Now put $m = \frac{3}{2}$ in equation of tangent $y = m x + (2 + m)$

We get

$2 y = 3 x + 7$

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Most popular questions from this chapter

In Problems 5鈥�24, graph each equation of the system. Then solve the system to find the points of intersection.

$x y = 1; y = 2 x + 1$

The ratio of a to b is $\frac{2}{3}$ . The sum of a and b is $10$ . What is the ratio of $a + b$ to $b - a$ ?

In Problems 5鈥�24, graph each equation of the system. Then solve the system to find the points of intersection.

$y = \sqrt{4 - x^{2}}; y = 2 x + 4$

In Problems 5鈥�24, graph each equation of the system. Then solve the system to find the points of intersection.

$x^{2} + y^{2} = 4; y^{2} - x = 4$

Graph each inequality by hand.

$x^{2} + y^{2} 鈮� 9$

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