TS EAMCET · Maths · Circle

If the line $x + y + 1 = 0$ intersects the circle $x^{2} + y^{2} + x + 3 y = 0$ at two points $A$ and $B,$ then the centre of the circle which passes through the points $A, B$ and the point of intersection of the tangents drawn at $A$ and $B$ to the given circle is

A $(\frac{5}{8}, \frac{5}{8})$
B $(1, - 1)$
C $(\frac{3}{4}, - \frac{1}{4})$
D $(3, - 4)$

Verified Solution

Answer & Solution

Correct Answer

(C) $(\frac{3}{4}, - \frac{1}{4})$

Step-by-step Solution

Detailed explanation

x2+y2+x+3y=0 Ab:x+y+1=0 is chord of contact from intersection point h,k AB; xh+yk+x+h2+3(y+k2)=0 ⇒x(h+12)+y(k+32)+h2+3k2=0 On comparing h+121=k+321=h2+3k21 h+12=k+32 h-k=1 ...(1) h+12=h2+3k2 h2-3k2=-12 h-3k=-1 ...(2) 1, 2…

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.

Same subject

Explore more questions on app

From TS EAMCET

Explore more questions on app

If the line x+y+1=0 intersects the circle x2+y2+x+3y=0 at two points A and B, then the centre of the circle which passes through the points A, B and the point of intersection of the tangents drawn at A and B to the given circle is

Step-by-step Solution

If the line $x + y + 1 = 0$ intersects the circle $x^{2} + y^{2} + x + 3 y = 0$ at two points $A$ and $B,$ then the centre of the circle which passes through the points $A, B$ and the point of intersection of the tangents drawn at $A$ and $B$ to the given circle is