Chapter 11: Q. 95 (page 775)

Use Descartes’s method to find the equation of the line tangent to the graph $2 x^{2} + 3 y^{2} = 14$ at the given point (1,2).

Short Answer

Expert verified

Equation of tangent is $y = - \frac{1}{3} x + \frac{7}{3}$

Step by step solution

Step 1. Given information

$2 x^{2} + 3 y^{2} = 14$ we have to find tangent at (1,2)

Step 2. General equation of tangent Now we put b=2-m y=mx +bWe get

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

$2 - m = b$

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

We get $y = m x + (2 - m)$

Step 3. Calculations

Now put $y = m x + (2 - m)$ in $2 x^{2} + 3 y^{2} = 14$

We get

$2 x^{2} + 3 {(m x + 2 - m)}^{2} = 14 2 x^{2} + 3 m^{2} x^{2} + 6 m x - 3 m^{2} x + 6 m x + 12 - 6 m - 3 m^{2} x - 6 m + 3 m^{2} = 14 (2 + 3 m^{2}) x^{2} + (12 m - 6 m^{2}) x + 3 m^{2} - 12 m - 2 - 0$

Step 3. Solving the quadratic equation

Now we have to solve the quadratic equation . we get

$x = \frac{(6 m^{2} - 12 m) \pm \sqrt{{(6 m^{2} - 12 m)}^{2} - 4 (2 + 3 m^{2}) (3 m^{2} - 12 m - 2)}}{2 (2 + 3 m^{2})}$

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

${(6 m^{2} - 12 m)}^{2} - 4 (2 + 3 m^{2}) (3 m^{2} - 12 m - 2) = 0 (3 m + 1)^{2} = 0 m = \frac{- 1}{3}$

Step 5. Equation of tangent

Now put $m = \frac{- 1}{3}$ in equation of tangent $y = m x + (2 - m)$

We get

$y = - \frac{1}{3} x + \frac{7}{3}$

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Use Descartes’s method to find the equation of the line tangent to the graph $2 x^{2} + 3 y^{2} = 14$ at the given point (1,2).

Short Answer

Step by step solution

Step 1. Given information

Step 2. General equation of tangent Now we put b=2-m y=mx +bWe get

Step 3. Calculations

Step 3. Solving the quadratic equation

Step 5. Equation of tangent

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Use Descartes’s method to find the equation of the line tangent to the graph 2x2+3y2=14at the given point (1,2).

Short Answer

Step by step solution

Step 1. Given information

Step 2. General equation of tangent Now we put b=2-m y=mx +bWe get

Step 3. Calculations

Step 3. Solving the quadratic equation

Step 5. Equation of tangent

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Use Descartes’s method to find the equation of the line tangent to the graph $2 x^{2} + 3 y^{2} = 14$ at the given point (1,2).