AP EAMCET · Maths · Circle

Let $\mathrm{S}$ be a circle concentric with the circle $3 x^2+3 y^2+x+y-1=0$. If the length of the tangent drawn from a point $(2,-2)$ to the given circle is the radius of the circle S, then the power of the point $(2,1)$ with respect to the circle $\mathrm{S}$ is

A $\frac{-137}{18}$
B $\frac{1}{18}$
C $\frac{-29}{18}$
D $\frac{23}{18}$

Verified Solution

Answer & Solution

Correct Answer

(C) $\frac{-29}{18}$

Step-by-step Solution

Detailed explanation

Given circle is $3 x^2+3 y^2+x+y-1=0$ \[ \begin{aligned} & \Rightarrow x^2+y^2+\frac{1}{3} x+\frac{1}{3} y-\frac{1}{3}=0 \\ & \therefore \text { Centre }=\left(-\frac{1}{6},-\frac{1}{6}\right), \text { radius }=\frac{\sqrt{14}}{6} \end{aligned} \] Length of tangents $P A$,…

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

Match the items of List-

I

with the items of List-

II

and choose the correct option

List- $I$		List- $II$
$(A)$	If $A$ is a non-singular matrix of order $3$ and $\| A \| = a,$ then $\|{(adj A^{- 1})}^{- 1}\| =$	$(I)$	null matrix
$(B)$	A is a non-singular matrix of order $3$ and $B$ is any matrix of order $3$ such that $A B = O$ , then $B$ is	$(II)$	$a^{2}$
$(C)$	$\|\begin{matrix} 1 & x & x^{2} \\ \cos (a - b) y & \cos α y & \cos (a + b) y \\ \sin (a - b) y & \sin α y & \sin (a + b) y \end{matrix}\|$ does not depend on	$(III)$	$b$
$(D)$	$A$ is a square matrix of order $3$ and $B = A - A^{T}$ , then $\|B\|$ is	$(IV)$	$a$
		$(V)$	$0$

The correct answer is

AP EAMCET 2019 Medium

Match the functions in Column I with their properties in Column II. In the following $[\mathrm{x}]$ denotes the greatest integer less than or equal to x

	Column I		Column II
A)	$x\|x\|$	I.	Strictly increasing and continuous in $(-1,1)$
B)	$\sqrt{\|x\|}$	II.	Continuous but not differentiable in $(-1,1)$
C)	$x+[x]$	III.	Differentiable in $(-1,1)$
D)	$\|x-1\|+\|x+1\|+\|x\|$	IV.	Differentiable in $(-1,0) \cup(0,1)$
		V.	Strictly increasing and not differentiable in $(-1,1)$

The correct match is

AP EAMCET 2025 Hard

If $\sec (x)=\cosh (\theta)$, then find $\tanh ^2\left(\frac{\theta}{2}\right)$

AP EAMCET 2020 Medium

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Let \(\mathrm{S}\) be a circle concentric with the circle \(3 x^2+3 y^2+x+y-1=0\). If the length of the tangent drawn from a point \((2,-2)\) to the given circle is the radius of the circle S, then the power of the point \((2,1)\) with respect to the circle \(\mathrm{S}\) is

Answer & Solution

Step-by-step Solution

See the Complete Solution

	Column I		Column II
A)	\(x\|x\|\)	I.	Strictly increasing and continuous in \((-1,1)\)
B)	\(\sqrt{\|x\|}\)	II.	Continuous but not differentiable in \((-1,1)\)
C)	\(x+[x]\)	III.	Differentiable in \((-1,1)\)
D)	\(\|x-1\|+\|x+1\|+\|x\|\)	IV.	Differentiable in \((-1,0) \cup(0,1)\)
		V.	Strictly increasing and not differentiable in \((-1,1)\)