TS EAMCET · Maths · Trigonometric Ratios & Identities

The combined equation of the two diameters of a circle which divide the circle into $4$ sectors is $a x^{2} + 2 h x y + b y^{2} = 0$ . If the area of the bigger sector is $5$ times the area of the smaller sector, then $\frac{|a + b|}{\sqrt{{(a - b)}^{2} + 4 h^{2}}} =$

A $\frac{1}{2}$
B $\frac{\sqrt{3}}{2}$
C $\frac{1}{\sqrt{2}}$
D $1$

Verified Solution

Answer & Solution

Correct Answer

(B) $\frac{\sqrt{3}}{2}$

Step-by-step Solution

Detailed explanation

Let angle between given pair of lines is θ according to given information. A2=5A1 12×π-θr2=12×θr2×5 π-θ=5θ θ=π6 ∵cosπ6=a+ba-b2+2h2 32=a+ba-b2+4h2 ∴a+ba-b2+4h2=32

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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The combined equation of the two diameters of a circle which divide the circle into 4 sectors is ax2+2hxy+by2=0. If the area of the bigger sector is 5 times the area of the smaller sector, then a+ba-b2+4h2=

Step-by-step Solution

The combined equation of the two diameters of a circle which divide the circle into $4$ sectors is $a x^{2} + 2 h x y + b y^{2} = 0$ . If the area of the bigger sector is $5$ times the area of the smaller sector, then $\frac{|a + b|}{\sqrt{{(a - b)}^{2} + 4 h^{2}}} =$