The circle $x^{2} + y^{2} + α x + β y + γ = 0$ is the image of the circle $x^{2} + y^{2} - 6 x - 10 y + 30 = 0$ across the line $3 x + y = 2$ . The value of $[α + β + γ]$ is (where $[\cdot]$ represents the floor function)

Answer (Detailed Solution Below)View answer ▾

Answer: Option C — 23

Detailed Solution

Given circle: centre $(3, 5)$ , radius $9 + 25 - 30 = 2$ .

Image of $(3, 5)$ in $3 x + y - 2 = 0$ : $k = \frac{2 ( 9 + 5 - 2 )}{10} = 2.4$ , image $= (3 - 3 k, 5 - k) = (- 4.2, 2.6)$ .

Image circle: $(x + 4.2)^{2} + (y - 2.6)^{2} = 4$ , so $α = 8.4$ , $β = - 5.2$ , $γ = 17.64 + 6.76 - 4 = 20.4$ .

$α + β + γ = 8.4 - 5.2 + 20.4 = 23.6$ , so $[α + β + γ] = 23$ .

✓ Check

The midpoint of

(3, 5)

and

(- 4.2, 2.6)

(- 0.6, 3.8)

, and

3 (- 0.6) + 3.8 = 2

: it lies on the mirror line.

⚠ Common trap

Rounding 23.6 to 24 is not the floor; the floor function takes 23.

Quick tip

$(x^{'}, y^{'}) = (x_{0}, y_{0}) - \frac{2 ( a x _{0} + b y _{0} + c )}{a ^{2} + b ^{2}} (a, b)$

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Circle — Preparation

This question tests your understanding of Circle, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

In coordinate geometry, a circle is the set of points at a fixed distance (the radius) from a fixed point (the centre). This topic covers the standard equation (x - h)^2 + (y - k)^2 = r^2, the general equation x^2 + y^2 + 2gx + 2fy + c = 0 with centre (-g, -f), the diameter form, and parametric equations. Exam questions also test the position of a point relative to a circle, the condition for a line to be a tangent, equations of tangents and normals, the length of a tangent from an external point, and the intersection of two circles. The fastest approach is often to compare the distance from the centre to a line with the radius: equal means tangent, smaller means the line cuts the circle. Practise the circle questions below, each with a step-by-step solution.

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The circle $x^{2} + y^{2} + α x + β y + γ = 0$ is the image of the circle $x^{2} + y^{2} - 6 x - 10 y + 30 = 0$ across the line $3 x + y = 2$ . The value of $[α + β + γ]$ is (where $[\cdot]$ represents the floor function)

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Circle — Preparation

The circle x2+y2+αx+βy+γ=0 is the image of the circle x2+y2−6x−10y+30=0 across the line 3x+y=2. The value of [α+β+γ] is (where [⋅] represents the floor function)

💬 Discussions on this question

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Circle — Preparation

The circle $x^{2} + y^{2} + α x + β y + γ = 0$ is the image of the circle $x^{2} + y^{2} - 6 x - 10 y + 30 = 0$ across the line $3 x + y = 2$ . The value of $[α + β + γ]$ is (where $[\cdot]$ represents the floor function)