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
${\left(-3\right)}^{2}=p\left(2\right)\phantom{\rule{0ex}{0ex}}9=2p\phantom{\rule{0ex}{0ex}}p=\frac{9}{2}$

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)
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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