If the circle x²+y²-2 g x+6 y-19 c=0, g, c ∈ ℝ passes through the point (6,1) and its centre lies on the line x-2 c y=8, then the length of intercept made by the circle on x-axis is :

Solution

Maths sir · 4 steps

Two conditions, two unknowns: find g and c first, then the x-intercept formula does the rest.

1. Read the centre. Compare with x²+y²+2Gx+2Fy+C=0. Here G=-g, F=3 and C=-19c, so the centre is (-G,-F)=(g, -3).
2. Use both conditions. Putting (6,1) in the circle gives 36+1-12g+6-19c=0, that is 12g+19c=43. Putting the centre (g,-3) in x-2cy=8 gives g+6c=8, so g=8-6c.
3. Solve. Substitute: 12(8-6c)+19c=43 gives 96-53c=43, so c=1 and g=2. The circle is x²+y²-4x+6y-19=0.

   x-intercept = 2√(G² - C) = 2√((-2)² - (-19)) = 2√23

4. So. The circle cuts a length of 2√23 on the x-axis, which is option (d).

The idea to keep. For x²+y²+2Gx+2Fy+C=0, the x-intercept is 2√(G²-C) and the y-intercept is 2√(F²-C). Watch the sign of C when you put it in.

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