Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line $4x + y = 16$

**OR**

Find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and length of latus-rectum of the ellips

Question

Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line $4x + y = 16$

**OR**

Find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and length of latus-rectum of the ellipse $\dfrac{x^2}{36} + \dfrac{y^2}{16} = 1$

Accepted Answer

>

> [!TLDR]
> The centre lies on both the perpendicular bisector of the chord through $(4,1)$, $(6,5)$ and the line $4x+y=16$; solving gives centre $(3,4)$ and radius$^2=10$.
>
> Since the circle passes through $(4,1)$ and $(6,5)$, its centre is equidistant from both points, so the centre lies on the **perpendicular bisector** of the segment joining them.
> …

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