If the points of intersection of the ellipses x 2 + 2 y 2 - 6 x - 12 y + 23 = 0 and 4 x 2 + 2 y 2 - 20 x - 12 y + 35 = 0 lie on a circle of radius r and centre (a,b), then the value of a b + 18 r 2 is. [2026]

Question

If the points of intersection of the ellipses $x^{2} + 2 y^{2} - 6 x - 12 y + 23 = 0$ and $4 x^{2} + 2 y^{2} - 20 x - 12 y + 35 = 0$ lie on a circle of radius $r$ and centre $(a, b)$ , then the value of $a b + 18 r^{2}$ is. [2026]

Smart Achievers · Accepted Answer

(4)

$By family of curve equation of circle will be$

$\Rightarrow S_{1} + λ S_{2} = 0$

$\Rightarrow (x^{2} + 2 y^{2} - 6 x - 12 y + 23) + λ (4 x^{2} + 2 y^{2} - 20 x - 12 y + 35) = 0$

$\Rightarrow for circle coeff of x^{2} = coeff of y^{2}$

$\Rightarrow λ = \frac{1}{2}$

$So equation of circle is$

$\Rightarrow x^{2} + y^{2} - \frac{16}{3} x - 6 y + \frac{27}{2} = 0$

$Centre (\frac{8}{3}, 3), Radius r = \sqrt{\frac{47}{18}}$

$∴$ $a b + 18 r^{2} = 8 + 47 = 55$

Solution

Q.
If the points of intersection of the ellipses $x^{2} + 2 y^{2} - 6 x - 12 y + 23 = 0$ and $4 x^{2} + 2 y^{2} - 20 x - 12 y + 35 = 0$ lie on a circle of radius $r$ and centre $(a, b)$ , then the value of $a b + 18 r^{2}$ is. [2026]

1 53

2 51

3 52

4 55

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