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Question 35 of 47

Q.For what values of a and b does the equation (a−2)x2+by2+(b−2)xy+4x+4y−1=0(a-2)x^2 + by^2 + (b-2)xy + 4x + 4y - 1 = 0 represents a circle ? Write down the resulting equation of the circle.

Puducherry TnboardTamil Nadu HSC First Year (DGE) Commerce Board 2023Subjective· 3mImportance★★★★★
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For a circle, coeff. of x2x^2 = coeff. of y2y^2 and coeff. of xy=0xy = 0. This gives b=2b = 2, a=4a = 4, and the circle 2x2+2y2+4x+4y−1=02x^2 + 2y^2 + 4x + 4y - 1 = 0.

Step 1 — Conditions for a circle. The general second-degree equation represents a circle when

(coeff. of x2)=(coeff. of y2)and(coeff. of xy)=0.\text{(coeff. of }x^2) = \text{(coeff. of }y^2) \quad\text{and}\quad \text{(coeff. of }xy) = 0.

Step 2 — Coefficient of xyxy is zero.

b−2=0  ⇒  b=2.b - 2 = 0 \;\Rightarrow\; b = 2.

Step 3 — Equal coefficients of x2x^2 and y2y^2.

Coeff. of x2=a−2x^2 = a - 2, coeff. of y2=by^2 = b. So

a−2=b=2  ⇒  a=4.a - 2 = b = 2 \;\Rightarrow\; a = 4.

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