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If the parabola y2 = 4x and x2 = 32y intersect at (16, 8) at an angle θ, then θ is equal to
- a)tan−1(3/5)
- b)tan−1(4/5)
- c)π
- d)π/2
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If the parabola y2= 4x and x2= 32yintersect at (16, 8) at an angle&the...
We can first find the coordinates of the point of intersection by substituting y^2 = 4x into x^2 = 32y:
x^2 = 32y
x^2 = 32(y^2/4)
x^2 = 8y^2
4x^2 = 32y^2
4x^2 = y^2
Substituting y^2 = 4x:
4x^2 = 4x
x^2 - x = 0
x(x-1) = 0
So x = 0 or x = 1. Since the parabolas intersect at (16,8), we know that x = 16. Substituting x = 16 into y^2 = 4x, we get y^2 = 64 and y = ±8. So the point of intersection is (16,8).
Now we need to find the slopes of the tangents to each parabola at this point. We can differentiate each equation with respect to x:
y^2 = 4x
2y dy/dx = 4
dy/dx = 2/y
x^2 = 32y
2x dx/dy = 32
dx/dy = 16/x
Substituting (16,8) into each equation, we get:
dy/dx = 2/8 = 1/4
dx/dy = 16/16 = 1
The angle between two lines with slopes m1 and m2 is given by:
θ = tan^-1(|(m1-m2)/(1+m1m2)|)
Substituting our slopes, we get:
θ = tan^-1(|(1/4 - 1)/(1 + 1/16)|)
θ = tan^-1(|(-3/4)/(17/16)|)
θ = tan^-1(|-12/17|)
So the angle between the tangents is approximately 35.5 degrees.
x^2 = 32y
x^2 = 32(y^2/4)
x^2 = 8y^2
4x^2 = 32y^2
4x^2 = y^2
Substituting y^2 = 4x:
4x^2 = 4x
x^2 - x = 0
x(x-1) = 0
So x = 0 or x = 1. Since the parabolas intersect at (16,8), we know that x = 16. Substituting x = 16 into y^2 = 4x, we get y^2 = 64 and y = ±8. So the point of intersection is (16,8).
Now we need to find the slopes of the tangents to each parabola at this point. We can differentiate each equation with respect to x:
y^2 = 4x
2y dy/dx = 4
dy/dx = 2/y
x^2 = 32y
2x dx/dy = 32
dx/dy = 16/x
Substituting (16,8) into each equation, we get:
dy/dx = 2/8 = 1/4
dx/dy = 16/16 = 1
The angle between two lines with slopes m1 and m2 is given by:
θ = tan^-1(|(m1-m2)/(1+m1m2)|)
Substituting our slopes, we get:
θ = tan^-1(|(1/4 - 1)/(1 + 1/16)|)
θ = tan^-1(|(-3/4)/(17/16)|)
θ = tan^-1(|-12/17|)
So the angle between the tangents is approximately 35.5 degrees.
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If the parabola y2= 4x and x2= 32yintersect at (16, 8) at an angle&the...
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If the parabola y2= 4x and x2= 32yintersect at (16, 8) at an angleθ, thenθis equal toa)tan−1(3/5)b)tan−1(4/5)c)πd)π/2Correct answer is option 'A'. Can you explain this answer?
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