The Circle — Flashcards
ICSE · Class 11 · Mathematics
28 flashcards for The Circle (ICSE Class 11 Mathematics) to test yourself on key terms and facts. Sample: "Solve: Find the equation of a circle"
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Solve: Find the equation of a circle with centre (0, 2) and radius 2.
Answer
Use the standard equation: (x-h)^2 + (y-k)^2 = r^2. Step 1: Substitute h = 0, k = 2, r = 2. Step 2: (x-0)^2 + (y-2)^2 = 2^2. Step 3: x^2 + (y-2)^2 = 4. Step 4: Expand: x^2 + y^2 - 4y = 0. Answer: x^2 …
Solve: Find the centre and radius of the circle (x + 5)^2 + (y - 3)^2 = 36.
Answer
Compare with (x-h)^2 + (y-k)^2 = r^2. Step 1: x + 5 means h = -5. Step 2: y - 3 means k = 3. Step 3: 36 = 6^2, so r = 6. Answer: Centre = (-5, 3), radius = 6.
Problem: Find the equation of a circle with centre (2, 2) passing through (4, 5).
Answer
Step 1: Radius is the distance between (2, 2) and (4, 5). CP = sqrt((4-2)^2 + (5-2)^2) = sqrt(4 + 9) = sqrt(13). Step 2: Use (x-h)^2 + (y-k)^2 = r^2. Step 3: (x-2)^2 + (y-2)^2 = 13. Step 4: Expand: x^…
Why is the radius always taken as a positive quantity in circle problems?
Answer
A radius is a length, and length cannot be negative. Step 1: The radius is the distance from the centre to a point on the circle. Step 2: Distance is always positive. Step 3: So even if an equation gi…
Solve: Find the centre and radius of x^2 + y^2 - 6x + 4y - 12 = 0.
Answer
Use x^2 + y^2 + 2gx + 2fy + c = 0. Step 1: Compare terms. 2g = -6, so g = -3. 2f = 4, so f = 2. Step 2: Centre = (-g, -f) = (3, -2). Step 3: Radius = sqrt(g^2 + f^2 - c) = sqrt((-3)^2 + 2^2 - (-12)).
When do you use the general equation x^2 + y^2 + 2gx + 2fy + c = 0?
Answer
Use it when the circle is given in expanded form or when the centre and radius are not directly visible. Worked example: For x^2 + y^2 - 8x + 10y - 12 = 0, 2g = -8, so g = -4. 2f = 10, so f = 5. Centr…
Problem: Check whether 9(x + 2)^2 + 25(y - 4)^2 = 225 represents a circle.
Answer
A circle in general form must have equal coefficients of x^2 and y^2, and no xy term. Step 1: Expand the idea. This equation has different coefficients for (x + 2)^2 and (y - 4)^2. Step 2: Since 9 and…
Why must the coefficients of x^2 and y^2 be equal in the general equation of a circle?
Answer
Because a circle has the same radius in every direction from its centre. Step 1: Unequal x^2 and y^2 coefficients stretch the graph differently in horizontal and vertical directions. Step 2: That chan…
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