The focus of the parabola is the centre of the circle C of radius 5. If the values of , for which C passes through the point of intersection of the lines 3x – y = 0 and x + y = 4, are and , then is equal to __________. [2025]

Question

The focus of the parabola $y^{2} = 4 x + 16$ is the centre of the circle C of radius 5. If the values of $λ$ , for which C passes through the point of intersection of the lines 3x – y = 0 and x + $λ$ y = 4, are $λ_{1}$ and $λ_{2}, λ_{1} < λ_{2}$ , then $12 λ_{1} + 29 λ_{2}$ is equal to __________. [2025]

Smart Achievers · Accepted Answer

(15)

We have, $y^{2} = 4 (x + 4)$

Focus of parabola = (–3, 0) $\Rightarrow$ Center $\equiv$ (–3, 0)

$∴$ Equation of circle is given by ${(x + 3)}^{2} + y^{2} = 25$

Intersection point of 3x – y = 0 and x + $λ$ y = 4 is

$(\frac{4}{3 λ + 1}, \frac{12}{3 λ + 1})$

Circle passes through the point of intersection of two lines 3x – y = 0 and x + $λ$ y = 4.

$\Rightarrow 144 λ^{2} + 24 λ - 168 = 0 \Rightarrow 18 λ^{2} + 3 λ - 21 = 0$

$\Rightarrow λ = - \frac{7}{6}, 1$

Now, $12 λ_{1} + 29 λ_{2}$ = –14 + 29 = 15.

Solution

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Solution

Q. The focus of the parabola y2=4x+16 is the centre of the circle C of radius 5. If the values of λ, for which C passes through the point of intersection of the lines 3x – y = 0 and x + λy = 4, are λ1 and λ2,λ1<λ2, then 12λ1+29λ2 is equal to __________. [2025]

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