The focus of the parabola $y^2=4 x+16$ is the centre of the circle $C$ of radius 5 . If the values of $\lambda$, for which C passes through the point of intersection of the lines $3 x-y=0$ and $x+\lambda y=4$, are $\lambda_1$ and $\lambda_2, \lambda_1 \lt \lambda_2$, then $12 \lambda_1+29…

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The focus of the parabola $y^2=4 x+16$ is the centre of the circle $C$ of radius 5 . If the values of $\lambda$, for which C passes through the point of intersection of the lines $3 x-y=0$ and $x+\lambda y=4$, are $\lambda_1$ and $\lambda_2, \lambda_1 \lt \lambda_2$, then $12 \lambda_1+29 \lambda_2$ is equal to ________

Aicharya · Accepted Answer

$y^2=4(x+4)$
Equation of circle
$(x+3)^2+y^2=25$
Passes through the point of intersection of two lines $3 x-y=0$ and $x+\lambda y=4$
$\left(\frac{4}{3 \lambda+1}, \frac{12}{3 \lambda+1}\right) \text {, we get }$
$\begin{aligned}
& \lambda=-\frac{7}{6}, 1 \\ & 12 \lambda_1+29 \lambda_2 -14+29=15
\end{aligned}$

The focus of the parabola $y^2=4 x+16$ is the centre of the circle $C$ of radius 5 . If the values of…

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