Show that the tangent at P(ct, c/t) on the curve x=ct, y=c/t has equation x+t2y=2ct. Show the area of…

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Show that the tangent at P(ct, c/t) on the curve x=ct, y=c/t has equation x+t2y=2ct. Show the area of triangle QOR is constant.

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Let's solve questions 7 and 8 from the image you sent. Questions 6 i) and 6 ii) were previously addressed (they were numbered as 5 i) and 5 ii) in the original context).

Question 7: Show that the tangent at the point P, with parameter $t$ , on the curve whose parametric equations are $x=ct$ , $y=\frac{c}{t}$ , where $c$ is a constant, has equation $x+t^2y = 2ct$ . This tangent meets the x-axis at the point Q and the line through P parallel to the x-axis cuts the y-axis at a point R. Show that the area of triangle QOR, where O is the origin, is a constant.

Find the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$ .
Given $x = ct$ , so $\frac{dx}{dt} = c$ .
Given $y = \frac{c}{t} = ct^{-1}$ , so $\frac{dy}{dt} = -ct^{-2} = -\frac{c}{t^2}$ .
Find the gradient of the tangent $\frac{dy}{dx}$ .
$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-c/t^2}{c} = -\frac{1}{t^2}$
Find the equation of the tangent at point P.
The coordinates of point P are $(x_P, y_P) = (ct, \frac{c}{t})$ .
Using the point-slope form $y - y_P = m(x - x_P)$ :
$y - \frac{c}{t} = -\frac{1}{t^2}(x - ct)$
Multiply the entire equation by $t^2$ to clear denominators:
$t^2y - t^2\left(\frac{c}{t}\right) = -(x - ct)$ $t^2y - ct = -x + ct$
Rearrange the terms to match the required form:
$x + t^2y = ct + ct$ $x + t^2y = 2ct$
This shows the equation of the tangent.
Find the coordinates of point Q.
Point Q is where the tangent meets the x-axis, so $y=0$ .
Substitute $y=0$ into the tangent equation $x+t^2y = 2ct$ :
$x + t^2(0) = 2ct$ $x = 2ct$
So, point Q is $(2ct, 0)$ .
Find the coordinates of point R.
Point R is where the line through P parallel to the x-axis cuts the y-axis.
A line through P parallel to the x-axis has the equation $y = y_P$ .
Since $P = (ct, \frac{c}{t})$ , the equation of this line is $y = \frac{c}{t}$ .
This line cuts the y-axis when $x=0$ . So, point R is $(0, \frac{c}{t})$ .
Show that the area of triangle QOR is a constant.
The vertices of the triangle are O $(0,0)$ , Q $(2ct, 0)$ , and R $(0, \frac{c}{t})$ .
This is a right-angled triangle with legs along the x and y axes.
The length of the base OQ is $|2ct|$ .
The length of the height OR is $|\frac{c}{t}|$ .
The area of triangle QOR is given by:
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
Assuming $c>0$ and $t>0$ (or considering absolute values for general $c, t$ ):
$\text{Area} = \frac{1}{2} \times (2ct) \times \left(\frac{c}{t}\right)$ $\text{Area} = \frac{1}{2} \times 2c^2$ $\text{Area} = c^2$
Since $c$ is a constant, $c^2$ is also a constant.
The area of triangle QOR is $\boxed{c^2}$ .

Question 8: Express $\frac{2}{(1+x)(1+3x)}$ in partial fractions. Hence solve the differential equation $\

Show that the tangent at P(ct, c/t) on the curve x=ct, y=c/t has equation x+t2y=2ct. Show the area of triangle QOR is constant.

Step-by-step solution

Find the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$ .

Find the gradient of the tangent $\frac{dy}{dx}$ .

Find the equation of the tangent at point P.

Find the coordinates of point Q.

Find the coordinates of point R.

Show that the area of triangle QOR is a constant.

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More Mathematics Questions

Show that the tangent at P(ct, c/t) on the curve x=ct, y=c/t has equation x+t2y=2ct. Show the area of triangle QOR is constant.

Step-by-step solution

Find the derivatives dxdt\frac{dx}{dt}dtdx​ and dydt\frac{dy}{dt}dtdy​.

Find the gradient of the tangent dydx\frac{dy}{dx}dxdy​.

Find the equation of the tangent at point P.

Find the coordinates of point Q.

Find the coordinates of point R.

Show that the area of triangle QOR is a constant.

Still stuck on this one?

More Mathematics Questions

Find the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$ .

Find the gradient of the tangent $\frac{dy}{dx}$ .