Let dy/dx = (ax - by + a)/(bx + cy + a), where a, b, c are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is :

Solution

Maths sir · 5 steps

Cross-multiply, spot y dx + x dy = d(xy), and integrate; the circle conditions then fix b and c.

1. Cross-multiply. Write it as (bx + cy + a) dy = (ax - by + a) dx. Collect the b terms together: (ax + a) dx - b(y dx + x dy) - (cy + a) dy = 0. Since y dx + x dy = d(xy), every term can be integrated directly.

   (a x²)/2 + a x - b xy - (c y²)/2 - a y = K

2. Make it a circle. A circle has no xy term, so b = 0. The coefficients of x² and y² must be equal, so a/2 = -c/2, giving c = -a. Dividing by a/2 gives x² + y² + 2x - 2y = k.
3. Use the point. Put in (2, 5): 4 + 25 + 4 - 10 = 23, so k = 23. The centre is (-1, 1) and the radius is √(1 + 1 + 23) = 5.
4. Distance to centre. From (11, 6) to (-1, 1) the distance is √(12² + 5²) = √169 = 13. This is more than 5, so the point lies outside the circle.
5. So. The shortest distance is 13 - 5 = 8, which is option (b).

The idea to keep. If a differential equation is a circle, integrate it, then remove the xy term and match the x², y² coefficients. From an outside point, shortest distance is distance to centre minus radius.

Answer from the NTA final answer key. Solution by PrepWiser.