Help me to solve problem in parabola.?

Question

1) If parabolas y^2 = 4x and x^2 = 32y intersect at (16, 8) at an angle theta then theta = ?

2) If (a, b) the midpoint of chord passing through vertex of parabola y^2 = 4x then
a) a^2 = 2b
b) 2a = b^2
c) a = 2b
d) 2a = b

3) The tangents from the origin to the parabola y^2+4=4x are inclined at
a) pi / 6
b) pi / 4
c) pi / 3
d) pi / 2
Please give explanations of your answers.

Answer 1

1) the angle will be the angle between the tangents at that point. differentiate each expression to get the slope of the tangents.

y^2 = 4x
y = 2x^1/2
d y = 1/2(2/x^1/2) = 1 /x^1/2 = slope of first

x^2 = 32y
y = x^2/32
dy = 2x/32 = x/16 =slope of second

plug in x =16 to get the numerical values of the slopes

first slope is 1/sqrt(16) = 1/4
second slope is 16/16 = 1

the slopes are numerically equal to the tangent of their angles
first angle = tan^-1( 1/4) and second angle = tan^-1(1)

the angle between the tangents is:

theta = tan^-1(1) -tan^-1(1/4)
theta = 45 - 14 = 31 degrees


2)


3)

Answer 2

1) If parabolas y^2 = 4x and x^2 = 32y intersect at (16, 8) at an angle theta then theta = ?

The parabolas intersect at the same angle as their tangents at the point of intersection. To find the slope of the tangents take the derivative.

For the first parabola:
y² = 4x
2y(dy/dx) = 4
dy/dx = 4/(2y) = 4/(2*8) = 4/16 = 1/4

For the second parabola:
x² = 32y
2x = 32(dy/dx)
dy/dx = 2x/32 = x/16 = 16/16 = 1

The slope is equal to the tangent of the angle.

First parabola
tanα = 1/4
α = arctan(1/4)

Second parabola
tanβ = 1
β = arctan(1) = π/4

θ = β - α
θ = π/4 - arctan(1/4) ≈ 0.5404195 radians ≈ 30.963757°
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2) If (a, b) the midpoint of chord passing through vertex of parabola y^2 = 4x then

Incomplete question. I only have one point of the chord. What are its endpoints or its direction? What else does it pass thru?

Can't answer without more information.
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3) The tangents from the origin to the parabola y^2+4=4x are inclined at

y² + 4 = 4x
y² = 4x - 4

Take the derivative.

2y(dy/dx) = 4
dy/dx = 4/(2y) = 2/y

y = (2/y)x
y² = 2x

Set the equations equal.

y² = 4x - 4 = 2x
2x = 4
x = 2
y² = 2x = 2*2 = 4
y = ± 2

dy/dx = ±2/y = ±2/2 = ±1

arctan(1) = π/4

The answer is: b) pi / 4.