Find the length of the portion of the tangent to the curve $\displaystyle x = acos^3 \theta $, $\displaystyle y=a sin^3 \theta $ intercepted between coordinate axes.
Hello zorro
I'll assume the question means 'Find, in terms of $\displaystyle \theta$, the length ...'. In which case, it's fairly straightforward. Find, in terms of $\displaystyle \theta$:
- $\displaystyle \frac{dx}{d\theta}$
- $\displaystyle \frac{dy}{d\theta}$
- Hence $\displaystyle \frac{dy}{dx}$
- The equation of the tangent at the point with parameter $\displaystyle \theta$
Then find the values of $\displaystyle x$ and $\displaystyle y$ where this tangent crosses the axes, by putting $\displaystyle y = 0$ and $\displaystyle x = 0$ in turn, in this equation.
Then use Pythagoras Theorem to find the required length.
Grandad
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i have got half of it can u tell me what to do next ..........
$\displaystyle
\frac{dx}{dy} = \frac{sin^2 \theta cos \theta}{- cos^2 \theta sin \theta}
$
$\displaystyle
= -tan \theta
$
f'(0) = - tan (0) = 0
there the slope of the tangent is 0
Now the eq of the tangent at pt $\displaystyle (x_0,y_0)$ is (0,0)
therefore eq is $\displaystyle (y - y_0) = m (x-x_0)$
ie (y-0)=0(x-0)
ie y=0
Hello zorro
Note that you have written $\displaystyle \frac{dx}{dy}$, instead of $\displaystyle \frac{dy}{dx}$. Also, perhaps you mis-read my post: I said to find the equation of the tangent at the point whose parameter is $\displaystyle \theta$, not $\displaystyle 0$.
So, using $\displaystyle \frac{dy}{dx} = -\tan \theta$, the equation of the tangent whose paramter is $\displaystyle \theta$ is:
$\displaystyle y- a\sin^3\theta = -\tan\theta(x-a\cos^3\theta)$
Meets the $\displaystyle x$-axis where $\displaystyle y = 0$, and
$\displaystyle - a\sin^3\theta = -\tan\theta(x-a\cos^3\theta)$
$\displaystyle \Rightarrow a \sin^2\theta\cos\theta=x-a\cos^3\theta$
$\displaystyle \Rightarrow x=a\cos\theta(\sin^2\theta+\cos^2\theta)=$ ?
Can you take it from there?
Grandad
Hello zorro
We have found that the tangent meets the $\displaystyle x$-axis where $\displaystyle x = a\cos\theta$.
In a similar way, it meets the $\displaystyle y$-axis where $\displaystyle x = 0$, and therefore:$\displaystyle y-a\sin^3\theta =-\tan\theta(0-a\cos^3\theta)$So we want the distance between the points $\displaystyle (0,a\sin\theta)$ and $\displaystyle (a\cos\theta, 0)$; which is:$\displaystyle =a\sin\theta\cos^2\theta$$\displaystyle \Rightarrow y=a\sin\theta(\sin^2\theta +\cos^2\theta)$$\displaystyle =a\sin\theta$$\displaystyle \sqrt{a^2\cos^2\theta +a^2\sin^2\theta}$Grandad$\displaystyle =a$