Equation of the circle whose centre is on the line x = 2y and which passes through the points (-1, 2) and
(3,-2) is
(1) x2 + y2 - 12x - 6y - 5 = 0
(2) x + 7 - 4x – 2y + 5 = 0
(3) x2 + y2 - 4x - 2y - 5 = 0
(4) x + y - 8x - 4y - 5 = 0
Answers
Answer:
Step-by-step explanation:
Revision Notes on Circle
The equation of a circle with its center at C(x0, y0) and radius r is: (x – x0)2 + (y – y0)2 = r2
If x0 = y0 = 0 (i.e. the centre of the circle is at origin) then equation of the circle reduce to x2 + y2 = r2.
If r = 0 then the circle represents a point or a point circle.
The equation x2 + y2 + 2gx + 2fy + c = 0 is the general equation of a circle with centre (–g, –f) and radius √(g2+f2-c).
Equation of the circle with points P(x1, y1) and Q(x2, y2) as extremities of a diameter is (x – x1) (x – x2) + (y – y1)(y – y2) = 0.
For general circle, the equation of the chord is x1x + y1y + g(x1 + x) + f(y1 +y) + c = 0
For circle x2 + y2 = a2, the equation of the chord is x1x + y1y = a2
The equation of the chord AB
(A ≡ (R cos α, R sin α); B ≡ (R cos β, R sin β)) of the circle x2 + y2 = R2 is given by x cos ((α + β )/2) + y sin ((α - β )/2) = a cos ((α - β )/2)
Chords are equidistant from the center of a circle if and only if they are equal in length.
Equal chords of a circle subtend equal angles at the center
Components of Circle
The angle subtended by an arc at the center id double the angle subtended by the same arc at the circumference of the circle.
Angle between the tangent and the radius is 90°.
Angles in the same segment are equal.
Angle in a semi-circle is 90°.
Two angles at the circumference subtended by the same arc are equal.
The below table describes the equations of circle according to changes in radii and centers:
The point P(x1, y1) lies outside, on, or inside a circle ?
S ≡ x2 + y2 + 2gx + 2fy + c = 0, according as S1 ≡ x12 + y12 + 2gx1 + 2fy1 + c > = or < 0.
The equation of the chord of the circle x2 + y2 + 2gx + 2fy +c=0 with M(x1, y1) as the midpoint of the chord is given by:
xx1 + yy1 + g(x + x1) + f(y + y1) = x12 + y12 + 2gx1 + 2fy1 i.e. T = S1
In case the radius and the central angle of a triangle are given, the length of the chord can be computed using the formula
Length of the chord = 2r sin (c/2), where ‘c’ is the central angle and ‘r’ is the radius
Circle with ‘r’ as radius and ‘c’ as the central angle
If a circle has two secants QR and ST, then
circle has two secants
If there is a circle which has one tangent and one secant, then the square of the tangent is equal to the product of the secant segment and its external segment.
If a radius or the diameter of a circle is perpendicular to a chord, then it divides the chord into two equal parts. The converse also holds true.
Hence, in the below figure, if OB is perpendicular to PQ, then then PA = AQ.