1) if parabola y²=px passes through point (2,-3),find the length of latus rectum .

2) find the value of p so that the equation x² +y² - 2px+ 4y - 12=0 may represent a circle of radius 5 units .

3) one end of the diameter of a circle x2+y2-6x+5y-7=0 os (7,-8).find the co-ordinates of other end

In the given question, we are given a parabola $y^2=px$ which passes through the point (2,-3). So, substituting x = 2 and y = -3 in the given equation we get
$$\begin{gathered} {\left(-3\right)}^2=p\left(2\right) \\ 9=2p \\ p=\frac{9}{2} \end{gathered}$$

Now, substituting $p=\frac{9}{2}$in the given parabola, we get
$y^2=\frac{9}{2}x$  ...(1)

Next, comparing equation (1) with the equation of the parabola $y^2=4ax$, we get
$4a=\frac{9}{2}$  ...(2)

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Now we know that the length of latus rectum is 4a. So, using equation (2)
Length of the latus rectum = 4a
  = $\frac{9}{2}$
Therefore, the length of the latus rectum is $\frac{9}{2}$

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