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Slide3 : C I R C L E Definition:
A circle is the locus of a Point which moves in such a way that its Distance from a fixed point is always Constant. The fixed point is called the Centre of the circle and the constant Distance is called the radius of the circle.
Slide4 : C I R C L E The equation of a circle
when the centre and radius are given : Notes CP = r => CP2 = r2 => (x-h)2 + (y-k)2 = r2 * If (h, k) = 0, X2 + y2 = r2
Slide5 : C I R C L E The equation of a circle
If the end points of a diameter are given : (x-x1) (x-x2) + (y-y1) (y-y2) = 0
Slide6 : C I R C L E The general equation of the circle : Notes * ax2 + by2 + 2hxy + 2gx + 2fy + c = 0
If a = b i.e. co.eff of x2 = co.eff. of y2,
h = 0 i.e. no xy term
Slide7 : C I R C L E Parametric equation of the circle x2 + y2 = r2 : x = r cos, y = r sin, 0 2
x = (r(1–t2))/(1+r2), y = 2rt/(1+r2),
- < t <
Slide8 : C I R C L E Equation of the tangent to a circle at a point (x1, y1) : Notes xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0 * If x2 + y2 = a2, xx1 + yy1 = a2 Definition: A tangent of a circle is a straight line which intersects (touches) the circle in exactly one point. Tangent
Slide9 : C I R C L E Length of the tangent to the circle from a point (x1, y1) : Notes *If P, On the circle, PT2 = 0 (PT is zero).
Outside the circle, PT2 > 0 (PT is real).
Inside the circle, PT2 < 0 (PT is img..).
Slide10 : C I R C L E The condition for the line
y = mx + c to be a tangent
to the circle x2 + y2 = a2 : Notes c2 = a2 (1 + m2)
Slide11 : C I R C L E Equation of the chord of contact of tangents from a point to the circle : xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
Slide12 : C I R C L E The two circles touch externally : C1C2 = r1 + r2 The distance between their centres equal to the sum of their radii.
Slide13 : C I R C L E The two circles touch internally : C1C2 = C1P – C2P = r1 + r2 The distance between their centres equal to the difference of their radii.
Slide14 : C I R C L E Condition for two circles cut orthogonally : 2g1g2 + 2f1f2 = c1 + c2 Definition: Two circles are said to be orthogonal if the tangent at their point of intersection are at right angles. Orthogonal circle
Slide15 : Introduction:
A straight line is the simplest geometrical curve. The different forms of equation of a straight line: Slope-intercept form:
y = mx + c where ‘m’ is the slope of the straight line and ‘c’ is the y intercept. Point-slope form:
y - y1 = m(x - x1) + c where ‘m’ is the slope and (x1, y1) is the given point.
Slide16 : Normal form:
Equation of a straight line in terms of the length of the perpendicular p from the origin to the line and the angle which the perpendicular makes with x-axis.
Slide17 : X cos + y sin = p
Slide18 : Parametric form:
If two variables, say x and y, are functions of a third variable, say ‘’, then the functions expressing x and y in terms of are called the parametric representations of x and y. The variable is called the parameter of the function.
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Slide20 : General form:
The equation ax + by + c = 0 will always represent a straight line.
Slide21 : Angle between two straight lines:
Slide22 : The condition for the three straight lines to be concurrent: Three straight lines,
a1x + b1x + c1 = 0
a2x + b2x + c2 = 0
a3x + b3x + c3 = 0
Slide23 : T h a n k Y o u