For any two nonzero real numbers a and b if this line xa+yb=1 is a tangent to the circle x2+y2=1, then which of the following is true?

(A) 1a,1b lies inside the circle

AP EAMCET · Maths · Circle

For any two nonzero real numbers $a$ and $b$ if this line $\frac{x}{a} + \frac{y}{b} = 1$ is a tangent to the circle $x^{2} + y^{2} = 1$ , then which of the following is true?

A $(\frac{1}{a}, \frac{1}{b})$ lies inside the circle
B $(a, b)$ lies inside the circle
C $(\frac{1}{a}, \frac{1}{b})$ lies on the circle
D $(a, b)$ lies on the circle

Verified Solution

Answer & Solution

Correct Answer

(C) $(\frac{1}{a}, \frac{1}{b})$ lies on the circle

Step-by-step Solution

Detailed explanation

Given, Equation of circle as x2+y2=1, Now assuming a point x1,y1 on circle we have equation of tangent at this point given by xx1+yy1=1 ...............1, Now given xa+yb=1 is tangent to the given circle so on comparing with equation 1 we get, x1=1a & y1=1b So…

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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