# calculus

posted by .

two curves are orthogonal at a point of intersection of their tangents at that point cross at right angles. Show that the curves 2x^2+3y^2=5 and y^2=x^3 are orthogonal at (1,1) and (1,-1). Use parametric mode to draw the curves and to show the tangent lines

• calculus -

By inspection, the curves intersect at (1,1) and (1,-1)

ellipse: 4x + 6yy' = 0
y' = -2x/3y
at (1,1) slope = -2/3
at (1,-1) slope = 2/3

semicubical parabola: 2yy' = 3x^2
y' = 3x/2y
at (1,1) slope = 3/2
at (1,-1) slope = -3/2

The slopes at the intersections are negative reciprocals; hence the curves are orthogonal.

## Similar Questions

dy/dx= (y^2 -1)/x 1. Give the general equation of the curves that satisfy this equation. 2. Show that the straight lines y=1 and y=-1 are also solutions 3. Do any of the curves you found in 1) intersect y=1?
2. ### Calculus - Orthogonal Trajectories

Find the orthogonal trajectories of the family of curves: y = k*(e^-x) --------------- so k = y/(e^-x) differentiating we get: 1 = -k(e^-x)*(dx/dy) 1/(dx/dy) = -k(e^-x) dy/dx = -k(e^-x)...substituting for k: dy/dx = -(y/(e^-x))*(e^-x) …
3. ### calculus

if the tangent of two intersecting circles, at their points of intersection are perpendicular, the circles are said to be orthogonal. Show that the circles x^2+y^2-6x+4y+2=0 and x^2+y^2+8x+2y-22=0 are orthogonal. find the equation …
4. ### calculus

Show that the curves (y= √2sinx) and (y=√2cosx) intersect at right angles at a certain point with 0<x<π/2
5. ### maths

determine the co-ordinate of the poin of intersection of the curves y=x*x and y*y=8x. sketch the two curves and find the area enclosed by the two curves.
6. ### Math

The curves y=sinx and y=cosx intersects twice on the interval (0,2pi). Find the area of the region bounded by the two curves between the points of intersection.
7. ### Math

The curves y=sinx and y=cosx intersects twice on the interval (0,2pi). Find the area of the region bounded by the two curves between the points of intersection.
8. ### Dr. D ram D.A.V. Public school

A parabola y²=4x cuts the circle with centre at (6,5) orthogonal then the possible points of intersection between the curves are?
9. ### calculus 2

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. (Round your answer to two decimal places.) y = 8x^2− 3x, …
10. ### calc 2

Find the orthogonal trajectories for the family of curves y=(kx)^6.

More Similar Questions