Distance Between Two Points

To solve this problem, we'll follow the steps to find the distance of point P from the chord AB.
1. **Circle Equation Analysis:** The circle equation given is . The center is and radius .
2. **Chord AB in Circle:** The chord AB has length 12. The perpendicular distance from the center C to the chord can be found using the formula .

3. **Intersection P of Tangents:** The tangents at A and B intersect at point P, the external point from which equal tangents PA and PB are drawn to the circle. The External Distance Theorem helps us here, stating that if two tangents are drawn from an external point to a circle, they are equal.
4. **Distance of Point P from AB:** This is a geometry problem, where the tangents and the chord form a right triangle. The tangents from P to A and B are longer than the perpendicular from P to AB. Use the fact that the length of the perpendicular from the center C to AB aids in calculating this perpendicular distance of P from AB.
5. **Calculation of Distance:** Here, 5 times the distance of P from the chord AB is asked for. Given the geometry, using precise calculations, it is found:

This matches the specified expected range.

The plane P passes through the intersection of the planes: The equation of P is: Substituting the equations of : Step 2: Determine λ Since P is perpendicular to the plane : 2x + y - z + 1 = 0, the normal vector of P is perpendicular to the normal vector of P_3. This gives: where Taking the dot product: Simplify: Solve for \lambda: Step 3: Equation of Plane P Substitute \lambda = -8 into equation (1): Step 4: Distance of Point (-7, 1, 1) from Plane P The distance d of a point (x_1, y_1, z_1) from a plane ax + by + cz + d = 0 is: Here, (x_1, y_1, z_1) = (-7, 1, 1), and the equation of P is -6x - 13y - 25z + 46 = 0. Substituting: Simplify: Step 5: Calculate d^2 Conclusion:

To solve this problem, we need to find the point of intersection of two lines and then calculate the distance between this intersection point and a given point .

1. First, identify the equations of both lines.
   • The first line passes through the point and is parallel to the line given by the direction ratios . Therefore, the parametric equation of the line is:
   • The second line is given by the ratios:
   • Thus, the parametric equation of the second line is:
2. The next step is to find the intersection, which requires solving the equations:
3. Solving these equations simultaneously:
   1. From the first equation:
      
      Thus,
   2. From the second equation:
      
      Thus,
   3. Equating both expressions for :
      
      Cross-multiplying gives
      Solving for , we get so .
   4. Substitute into or to find .
      Using the first: (check consistency with second gives same result).
4. Calculate coordinates of intersection point using either parametric equation with or .
   • For the first line:
5. Now, compute the distance between points and using the distance formula:
   
   = .

Therefore, the distance of point from is .