AP EAMCET · Maths · Circle

If the circle $x^{2} + y^{2} - 4 x - 8 y - 5 = 0$ intersects the line $3 x - 4 y - m = 0$ in two distinct points, then the number of integral values of ' $m$ ' is

A $52$
B $51$
C $50$
D $49$

Verified Solution

Answer & Solution

Correct Answer

(D) $49$

Step-by-step Solution

Detailed explanation

If the circle is intersecting line 3x-4y=m at two distinct points then we can say that Radius > length of perpendicular from center to the given line Perpendicular distance, P=ax1+by1+ca2+b2, r=g2+f2-c ⇒4+16+5>32-44-m9+16 ⇒m+10<25…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If the circle x2+y2-4x-8y-5=0 intersects the line 3 x-4 y-m=0 in two distinct points, then the number of integral values of ' m ' is

Step-by-step Solution

If the circle $x^{2} + y^{2} - 4 x - 8 y - 5 = 0$ intersects the line $3 x - 4 y - m = 0$ in two distinct points, then the number of integral values of ' $m$ ' is