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1Exercise 11.1
Find the equation of the circle with centre (3, 4) and radius 5.
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(x - h)² + (y - k)² = r²
(x - 3)² + (y - 4)² = 25
x² - 6x + 9 + y² - 8y + 16 = 25
x² + y² - 6x - 8y = 0
2Exercise 11.1
Find the equation of the parabola with focus (2, 0) and directrix x = -2.
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Focus (a, 0) and directrix x = -a gives y² = 4ax with vertex at origin.
Here a = 2, so the parabola opens to the right:
y² = 4(2)x = 8x
3Exercise 11.1
Find the coordinates of the focus, axis, and equation of directrix for the parabola y² = 16x.
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Standard form y² = 4ax ⇒ 4a = 16 ⇒ a = 4.
Focus = (a, 0) = (4, 0)
Axis = x-axis (y = 0)
Directrix: x = -a = -4
4Exercise 11.1
Find the equation of the ellipse with foci (±4, 0) and length of major axis 10.
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Major axis along x-axis, length 2a = 10 ⇒ a = 5.
c = 4 (distance of focus from centre).
b² = a² - c² = 25 - 16 = 9 ⇒ b = 3.
Equation: x²/25 + y²/9 = 1
5Exercise 11.1
Find the equation of the hyperbola with vertices (±5, 0) and foci (±6, 0).
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a = 5, c = 6.
b² = c² - a² = 36 - 25 = 11.
Equation: x²/25 - y²/11 = 1
6Exercise 11.1
Find the centre and radius of the circle x² + y² + 8x - 6y = 0.
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Complete the square:
x² + 8x + y² - 6y = 0
(x + 4)² - 16 + (y - 3)² - 9 = 0
(x + 4)² + (y - 3)² = 25
Centre = (-4, 3), Radius = 5