Topic Details (Notes format)

How to Complete the Square for Conic Sections

Subject: Mathematics

Book: Maths Mastery

To rewrite conics in standard form, group x and y terms, complete the square for each. For instance, x² + 6x + y² – 2y= 6 transforms into (x²+6x+9)+(y²–2y+1)=6+9+1 → (x+3)²+(y–1)²=16, a circle. This systematically reveals center, radius (circle), or center, axes (ellipse, hyperbola). Completing the square is essential for identifying conic type, orientation, and focal properties. Mastery ensures quick conversions from general to standard forms, enabling geometric interpretations or direct application in advanced designs and analyses.

Practice Questions

A number is increased by 20% and then decreased by 10%. What is the net change?

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If x = 3 and y = 4, what is the value of x^2 + y^2?

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If 2x = 16, what is the value of x?

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The base of a triangle is 10 cm and its height is 6 cm. What is its area?

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If 2a + b = 10 and a - b = 4, what is the value of a?

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The perimeter of a rectangle is 40 cm, and its length is 12 cm. What is its width?

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If the sum of three consecutive integers is 72, what are the integers?

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If a = 4 and b = 5, what is the value of (a+b)^2?

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If a:b = 2:3 and b:c = 4:5, what is a:c?

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What is the square root of 0.25?

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