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STEP III 2007 question 4 solutionTSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP III 2007 question 4 solution
As required. Tangent to curve at P is
y = 0
As required. Next part.
Right so
As required. Normal to P
You really need to look at a diagram here. If CQ is parallel to y axis the equation Let
Now We are told that the distance between P and C is rho = a cot t
So if the coordinates of C are The distance between P and C is rho. Which we are told it is in the question. This means that the coordinates of C are Solution by insparato. |
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![\displaystyle \rho = \frac{ [(-a\sin t + \frac{a}{\sin t})^2 + a^2\cos^2t]^{\frac{3}{2}}}{|[(-a\sin t + \frac{a}{\sin t}).(-a\sin t) -a\cos t.(-a\cos t - a\cot t\cosect)]|}](http://www.thestudentroom.co.uk/latexrender/pictures/55/550f7ea13f0ac8a0ea24dfede41c8ac4.png "\displaystyle \rho = \frac{ [(-a\sin t + \frac{a}{\sin t})^2 + a^2\cos^2t]^{\frac{3}{2}}}{|[(-a\sin t + \frac{a}{\sin t}).(-a\sin t) -a\cos t.(-a\cos t - a\cot t\cosect)]|}")
![\displaystyle \rho = \frac{[a^2\sin ^2t + \frac{a^2}{\sin ^2t} - 2a^2 + a^2\cos^2t]^{|\frac{3}{2}}}{|a^2\sin ^2t-a^2 + a^2\cos^2t + a^2 \cot ^2t|}](http://www.thestudentroom.co.uk/latexrender/pictures/9b/9b8ca0ee506b344458d7a3186309d037.png "\displaystyle \rho = \frac{[a^2\sin ^2t + \frac{a^2}{\sin ^2t} - 2a^2 + a^2\cos^2t]^{|\frac{3}{2}}}{|a^2\sin ^2t-a^2 + a^2\cos^2t + a^2 \cot ^2t|}")
![\displaystyle \rho = \frac{[\frac{a^2-a^2\sin ^2t}{\sin ^2t}]^{\frac{3}{2}}}{a^2\cot ^2t}](http://www.thestudentroom.co.uk/latexrender/pictures/a1/a13558deb1afe2e23c0b201fd7b340fd.png "\displaystyle \rho = \frac{[\frac{a^2-a^2\sin ^2t}{\sin ^2t}]^{\frac{3}{2}}}{a^2\cot ^2t}")
![\displaystyle \rho = \frac{[a^2\cot ^2t]^{\frac{3}{2}}}{a^2\cot ^2t}](http://www.thestudentroom.co.uk/latexrender/pictures/ac/ac8c70e065aa3f2a543aae6fcc7df76b.png "\displaystyle \rho = \frac{[a^2\cot ^2t]^{\frac{3}{2}}}{a^2\cot ^2t}")
![\displaystyle \text{Distance}^2 = a^2\cos^2t + [(\frac{a-a\sin ^2t}{\sin t}]^2](http://www.thestudentroom.co.uk/latexrender/pictures/e0/e013c306e7eca276dc15866299bbb343.png "\displaystyle \text{Distance}^2 = a^2\cos^2t + [(\frac{a-a\sin ^2t}{\sin t}]^2")
![\displaystyle \text{Distance}^2 = a^2\cos^2t + [\frac{a\cos^2t}{\sin t}]^2](http://www.thestudentroom.co.uk/latexrender/pictures/1e/1eacc775787fbfc2f39647bcfab2d2a4.png "\displaystyle \text{Distance}^2 = a^2\cos^2t + [\frac{a\cos^2t}{\sin t}]^2")
