Q.If P is a point on the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{25} = 1$ whose foci are S and S$'$, then $PS + PS' = 8$.

The key idea is that for any point on an ellipse, the sum of distances to the two foci is constant and equals $2a$, where $a$ is the semi-major axis length.

Step 1: The given ellipse is $\frac{x^2}{16} + \frac{y^2}{25} = 1$. Here $a^2 = 25$ (since $25 > 16$, the major axis is vertical), so $a = 5$.

Step 2: The constant sum property states $PS + PS' = 2a$. …

1. Understanding the standard form of an ellipse

An ellipse is defined as the set of all points $P$ such that the sum of distances to two fixed points (the foci $S$ and $S'$) is constant. That constant is $2a$, where $a$ is the semi-major axis length.

The given equation is:

Here, the denominator under $y^2$ is larger ($25 > 16$), so the major axis is vertical (along the $y$-axis). For an ellipse of the form $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ with $a > b$, we have:

• Semi-major axis: $a = \sqrt{25} = 5$
• Semi-minor axis: $b = \sqrt{16} = 4$

So here, $2a = 2 \times 5 = 10$.

2. Why the statement $PS + PS' = 8$ is wrong

The problem claims that $PS + PS' = 8$. But from the definition of the ellipse, the sum must be $10$. The number $8$ is actually $2b$ (twice the semi-minor axis), which is a common confusion — students sometimes mix up $a$ and $b$ when the major axis is vertical. …