Orthogonal Intersection of Two Circles

Orthogonal Intersection of Two Circles:

1. Orthogonal Intersection Definition:

  • Two circles intersect orthogonally when their intersection points form right angles (90 degrees).

2. Equation of Circles:

  • Circles are defined by equations (xh1)2+(yk1)2=r12 and (xh2)2+(yk2)2=r22 with centers (h1,k1) and (h2,k2), and radii r1 and r2 respectively.

3. Conditions for Orthogonal Intersection:

  • Perpendicular Tangents: At the point of intersection, the tangents to each circle are perpendicular to each other.
  • Product of Slopes: The product of the slopes of the tangents drawn from the point of intersection to each circle is equal to -1.

4. Calculating Orthogonal Intersection:

  • Step 1: Determine Tangents' Slopes:
    • Calculate the slopes of the tangents by finding the derivatives of the circle equations at the points of intersection.
  • Step 2: Check for Perpendicularity:
    • Verify that the product of slopes is -1 to confirm orthogonal intersection.

5. Example:

Consider the following circles:

C1:(x2)2+(y3)2=4 with center (2,3) and radius r1=2.

C2:(x+1)2+(y1)2=9 with center (1,1) and radius r2=3.

Steps to Determine Orthogonal Intersection:

Step 1: Find the Tangents' Slopes

Let's differentiate the circle equations to find the slopes of the tangents at the point of intersection.

For C1:(x2)2+(y3)2=4:

  • Differentiating implicitly, we get dydx=x2y3.
  • At the point of intersection (x0,y0), the slope of the tangent for C1 will be dydx evaluated at (x0,y0).

For C2:(x+1)2+(y1)2=9:

  • Similarly differentiating, we get dydx=x+1y1.
  • At the point of intersection (x0,y0), the slope of the tangent for C2 will be dydx evaluated at (x0,y0).

Step 2: Check for Perpendicularity

Evaluate the slopes at the point of intersection and confirm that the product of slopes is -1 to verify orthogonal intersection.

Calculation:

Let's assume the circles intersect at point (x0,y0).

For C1 at (x0,y0) (C1:(x2)2+(y3)2=4:

  • Calculate dydx at (x0,y0): dydx=x02y03.

For C2 at (x0,y0) (C2:(x+1)2+(y1)2=9:

  • Calculate dydx at (x0,y0): dydx=x0+1y01.

Check if the product of slopes is -1: (x02y03)×(x0+1y01)=1.

Conclusion:

If the product of slopes is indeed -1 for the given values of x0 and y0, it confirms that the tangents to the circles at their intersection point are perpendicular. Hence, the circles intersect orthogonally at that point.