Learn with Amar
Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Read semiaxes from squared denominators.
- Locate foci on the major axis.
- Check the distance-sum definition.
Before you start
Square roots, coordinate distance, and substituting into equations.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
An oval diagram follows x²/25 + y²/9 = 1, with coordinates in centimetres. Find its axis lengths, foci, and eccentricity; check the point (0, 3).
Why this math matters
An ellipse is a precise geometric curve, not just anything oval. Its equation and two-focus description give independent ways to verify a plotted outline.
Set up the model
A useful answer starts with clear assumptions:
- The ellipse is centred at the origin with axes aligned to the coordinate axes.
- The drawing uses an equal scale in both directions.
- The curve is nondegenerate: both squared semiaxes are positive.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Read an ellipse from its equation and foci
PausedQuestion: Start with the question. Paused.
Question
Start with the question
An oval diagram follows x²/25 + y²/9 = 1, with coordinates in centimetres. Find its axis lengths, foci, and eccentricity; check the point (0, 3).
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Read the semiaxes
a = √25 = 5 cm; b = √9 = 3 cm
The larger denominator lies under x², so the major axis is horizontal. Full axis lengths are 10 cm and 6 cm.
Find the focal distance
c = √(a² − b²) = √16 = 4 cm; e = c/a = 4/5
The foci are (−4, 0) and (4, 0). Their separation is 8 cm; c measures only centre-to-focus distance.
Check a boundary point
0/25 + 9/9 = 1; √(4² + 3²) + √(4² + 3²) = 10 cm
Point (0, 3) satisfies the equation. Its distances to the two foci also sum to 2a, confirming the geometric description.
The result
Axis lengths are 10 cm and 6 cm; foci are (±4, 0); eccentricity is 0.8.
The foci lie inside the ellipse. Moving them closer together while keeping a fixed makes the ellipse more nearly circular.
Common mistakes to catch
- Denominators give squared semiaxes, not full axis lengths.
- For an ellipse, c² = a² − b², not a² + b².
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
For x²/169 + y²/25 = 1, locate the foci.
Show a hint
Compute √(169 − 25).
Reveal answer and explanation
(−12, 0) and (12, 0)
The horizontal semimajor axis is 13 and c = 12.
Practice 2
Is (3, 12/5) on x²/25 + y²/9 = 1?
Show a hint
Substitute exactly.
Reveal answer and explanation
Yes
9/25 + (144/25)/9 = 9/25 + 16/25 = 1.
Take the idea with you
Use both algebraic substitution and geometric meaning when checking a conic drawing.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Find the part of a displacement along a chosen direction
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