What is the name of the curve where its radial vector ....

In summary, the curve in question is one where its radial vector, represented by y=mx, intersects the tangent line, represented by y=m1x+c1, at every point with an angle of 135 degrees or a=3pi/4. To find the general equation of this curve, one can use the point of intersection (x1,y1) and solve for m and m1 in terms of x1, y1, and c1. Then, using the formula for the angle between two lines, tan(a)=m2-m1/(1+m2m1), an equation can be derived in terms of x1, y1, and c1 to represent the curve. There may be shorter methods to find
  • #1
yoyo
21
0
what is the name of the curve where its radial vector drawn from the origin intersects the curves tangent line at a=3pi/4 or 135 degree at every point?

can anybody even show me how the curve looks like or what is the general eqaution of this curve?

thanks
 
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  • #2
Take the genaral equation of the radial vector as y=mx and tangent line as y= m1x + c1. Let the point of intersection be (x1, y1). This means both the above equations should be satisefied by (x1,y1). So plug in x1, y1 and get m and m1 in terms of x1, y1 and c1. Now use the fact that the angle 'a' between two lines is given by tan (a) = m2 - m1 / (1 + m2m1). You will get an expression in terms of x1, y1 and c1. This is the equation of the curve. Should look familiar.

Note: This may be a long way. There may be short cuts. I can't think of any right now. :zzz:
 
  • #3


The curve you are describing is called a cardioid. It gets its name from the Latin word for heart, "cardia", because its shape resembles a heart with a cusp at the top. The general equation for a cardioid is r = a(1 + cosθ), where r is the radial distance from the origin and θ is the angle measured counterclockwise from the positive x-axis. When a = 3π/4 or 135 degrees, the radial vector intersects the tangent line at every point, creating the distinctive heart shape. You can visualize this curve by plotting points with different values of θ and r on a polar coordinate system.
 

1. What is the name of the curve where its radial vector is always perpendicular to its tangent vector?

The curve is called a circle, and it is a type of circular curve with a constant radius.

2. Is the curve where its radial vector is always perpendicular to its tangent vector always circular?

No, the curve can also be an ellipse or hyperbola, depending on the relative magnitudes of the radial and tangent vectors.

3. Can the curve where its radial vector is always perpendicular to its tangent vector have varying radius?

Yes, the curve can have a varying radius and is then known as a spiral.

4. What is the significance of the radial and tangent vectors in this type of curve?

The radial vector represents the distance from the center of the curve to any point on the curve, while the tangent vector represents the direction of motion at that point.

5. Are there any real-life examples of this type of curve?

Yes, the motion of planets around the sun can be approximated by circular, elliptical, or hyperbolic curves where the radial vector is always perpendicular to the tangent vector.

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