A-Level AQA Maths Pure Implicit Differentiation: A curve, C, passes through the p

A curve, C, passes through the point with coordinates (1, 6) The gradient of C is given by $$\frac{dy}{dx} = \frac{1}{6}(xy)^2$$ Show that C intersects the coordinate axes at exactly one point and state the coordinates of this point - AQA - A-Level Maths Pure - Question 11 - 2021 - Paper 1

Question 11

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A curve, C, passes through the point with coordinates (1, 6) The gradient of C is given by $$\frac{dy}{dx} = \frac{1}{6}(xy)^2$$ Show that C intersects the coordi... show full transcript

Worked Solution & Example Answer:A curve, C, passes through the point with coordinates (1, 6) The gradient of C is given by $$\frac{dy}{dx} = \frac{1}{6}(xy)^2$$ Show that C intersects the coordinate axes at exactly one point and state the coordinates of this point - AQA - A-Level Maths Pure - Question 11 - 2021 - Paper 1

Step 1

Separate Variables and Integrate

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Answer

To find the equation of the curve, we start by separating the variables:

$\frac{dy}{dx} = \frac{1}{6}(xy)^2$

Rearranging gives:

$\frac{dy}{(y)^2} = \frac{1}{6} x dx$

Now, we integrate both sides. The left side becomes:

$\int \frac{dy}{y^2} = -\frac{1}{y}$

And the right side is:

$\int \frac{1}{6} x dx = \frac{1}{12} x^2 + C$

Thus, we have:

$-\frac{1}{y} = \frac{1}{12} x^2 + C$

Step 2

Determine the Constant of Integration

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Answer

Next, we substitute the point (1, 6) to find the constant C:

$-\frac{1}{6} = \frac{1}{12} (1)^2 + C$

This simplifies to:

$-\frac{1}{6} = \frac{1}{12} + C\n\Rightarrow C = -\frac{1}{6} - \frac{1}{12} = -\frac{2}{12} - \frac{1}{12} = -\frac{3}{12} = -\frac{1}{4}$

Thus, the equation is:

$-\frac{1}{y} = \frac{1}{12} x^2 - \frac{1}{4}$

Step 3

Find the y-intercept

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Answer

To find the y-intercept, set x = 0:

\Rightarrow y = 4$$ So, the y-intercept is (0, 4).

Step 4

Determine x-intercept

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Answer

Next, to find the x-intercept, we set y = 0:

However, since we have:

$-\frac{1}{y} = \frac{1}{12} x^2 - \frac{1}{4}$

The term $-\frac{1}{y}$ is undefined when y = 0, which indicates that the curve does not intersect the x-axis.

Hence, C does not intersect the x-axis.

Step 5

Conclusion

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Answer

The curve intersects the coordinate axes at exactly one point, which is the y-intercept (0, 4). It does not intersect the x-axis.

A curve, C, passes through the point with coordinates (1, 6) The gradient of C is given by $$\frac{dy}{dx} = \frac{1}{6}(xy)^2$$ Show that C intersects the coordinate axes at exactly one point and state the coordinates of this point - AQA - A-Level Maths Pure - Question 11 - 2021 - Paper 1

Separate Variables and Integrate

Determine the Constant of Integration

Find the y-intercept

Determine x-intercept

Conclusion

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