Question

what is the focal distance of of point in parabola x square =py( 12,16) ?
please help me friends

Answer 1

Hey,

Here is your answer ,

The parabola x2=pyx2=py passes through the point (12,16) [given]. It means that the point (12,16) must satisfy the equation of the parabola x2=py

So x2=pyx2=py

122=p(16)122=p(16)

144=16p144=16p

P=9P=9

So the equation of the parabola is

x2=9yx2=9y

x2=4(94)yx2=4(94)y

Now let's understand the equation of the type x2=4ayx2=4ay


Now comparing x2=4ayx2=4ay with our equation x2=4(94)yx2=4(94)y we get a=94a=94

And we know that focus of the equation of the type x2=4ayx2=4ay is (0,a)(0,a)

So coordinate of the focus of the equation x2=4(94)yx2=4(94)y is (0,94)(0,94)

The focal distance of the point is the distance between the focus and the given point on the parabola.

So focal distance of the point (12,16)(12,16) on the parabola is given by distance between (0,94)(0,94)and (12,16)(12,16)

Now the distance between any two point (x1,x2)(x1,x2) and (y1,y2)(y1,y2) is given by

d=(x1−x2)2+(y1−y2)2−−−−−−−−−−−−−−−−−−√d=(x1−x2)2+(y1−y2)2

So the distance between the point (0,94)(0,94)and (12,16)(12,16) is given by

d=(0−12)2+(94−16)2−−−−−−−−−−−−−−−−−√d=(0−12)2+(94−16)2

d=144+(302516)−−−−−−−−−−√d=144+(302516)

d=532916−−−−√d=532916

d=734.

Hence proved.

Happy to answer you.

@M.S.V.

Answer 2

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