TS EAMCET · Maths · Circle

The line $y=m x+c$ intercepts the circle $x^2+y^2=r^2$ in two distinct points, if

A $-r \sqrt{1+m^2} < c < r \sqrt{1+m^2}$
B $c < -r \sqrt{1+m^2}$
C $c < r \sqrt{1+m^2}$
D None of the above

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Answer & Solution

Correct Answer

(A) $-r \sqrt{1+m^2} < c < r \sqrt{1+m^2}$

Step-by-step Solution

Detailed explanation

Equation of the circle is \[ x^2+y^2=r^2 \] and the line is $m x-y+c=0$ The line (ii) intersect (i) in two distinct points, if the length of the $\perp^r$ from the centre $(0,0)$ to the line (ii) is less then $r$ Thus, $\left|\frac{0-0+c}{\sqrt{m^2+1}}\right| < r$…

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

Seven scientists $S_1, S_2, \ldots . ., S_7$ are invited to deliver one lecture each in a conference. The number of ways all the seven lectures can be arranged such that the lecture of $S_1$ is prior to that of $S_3$ and the lecture of $S_3$ is prior to that of $S_7$ is

TS EAMCET 2021 Easy

Match the items of List-I with those of the items of List-II

	List- $I$		List- $II$
A	Range of $\sec^{- 1} [1 + \cos^{2} x]$ , $[\cdot]$ denote greatest integer function	I	odd function
B	Domain of $f (x)$ where $f (x + \frac{1}{x}) = x^{2} + \frac{1}{x^{2}}$	II	$\{0, \frac{1}{2}\}$
C	$f (x + y) = f (x) + f (y); f (1) = 5$	III	$\{\sec^{- 1} 5, \sec^{- 1} 4\}$
D	$\sin^{- 1} x - \cos^{- 1} x + \sin^{- 1} (1 - x) = 0 \Rightarrow x \in$	IV	$R$
		V	$\{\sec^{- 1} 1, \sec^{- 1} 2\}$

TS EAMCET 2020 Hard

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The line \(y=m x+c\) intercepts the circle \(x^2+y^2=r^2\) in two distinct points, if

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