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The Circle — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSThe Circle4 Marks
Question
Find the equation of the circle which passes through the points $(2, 3)$ and $(4,5)$ and the centre lies on the straight line $y - 4x + 3 = 0$
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Answer

Find the equation of the circle which passes through the point (2, 3) and ( 4, 5) and the centre lies on the straight line y - 4x + 3 = 0. Let the equation of required circle be $x^2 + y^2 + 2gx+ 2fy + c = 0$ which passes through the point (2, 3) and ( 4, 5) .
$\therefore$ 13 + 4g + 6f + c = 0 ......... (1) 41 + 8g +10f + c = 0 ........... (2) Centre (-g, -f) lies on y - 4x + 3 = 0 -f + 4g = -3 .......... (3) Subtracting (1) from (2), we get 28 + 4g + 4f = 0 ........ (4) Solving (3) and (4)
we get, f = -5 and g = -2 Substituting values off and g in (2) we get, 41 - 16 - 50 + c = 0 c = 25 $\therefore$ The required equation of the circle is, $x^2 + y^2 - 4x - 10y + 25 = 0$.

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