Answer: Given, x2 + y2 β 6x + 5y β 7 = 0 Centre (3, -5/2) ( - 1, 3) & (????, Ξ²) are the 2 extremities of the diameter Using mid - point formula, $\begin{array}{l} \frac{{\alphaΒ - 1}}{2} = 3\\...

### Show that the quadrilateral formed by the straight lines x β y = 0, 3x + 2y = 5, x β y = 10 and 2x + 3y = 0 is cyclic and hence find the equation of the circle.

Answer: Slope of CD = 1 AB||CD Slope of BD = AC = - 1 AC||B They form a rectangle with all sides = 900 The quadrilateral is cyclic as sum of opposite angles...

### Find the equation of the circle circumscribing the triangle formed by the lines x + y = 6, 2x + y = 4 and x + 2y = 5.

Answer: The required circle equation is, Using Laplace Expansion, 27(x2 + y2) - 459x - 513y + 1350 = 0 x2 + y2 - 17x - 19 + 50 =...

### Find the equation of a circle passing through the origin and intercepting lengths a and b on the axes.

Answer: AD = b units and AE = a units. D(0, b), E(a, 0) and A(0, 0) lies on the circle. C is the centre. The general equation of a circle: (x - h)2 + (y - k)2 = r2...

### Find the equation of the circle which passes through the points A(1, 1) and B(2, 2) and whose radius is 1. Show that there are two such circles.

Answer: The general equation of a circle is, (x - h)2 + (y - k)2 = r2 β¦(i) (h, k) is the centre and r is the radius. Putting A(1, 1) in (i) (1 - h)2 + (1 - k)2 = 12 ...

### Prove that the centres of the three circles ,Β andΒ are collinear.

Answer: Given, x2 + y2 β 4x β 6y β 12 = 0 Centre ( - g1, - f1) = (2, 3) x2 + y2 + 2x + 4y β 5 = 0 Centre ( - g2, - f2) = ( - 1, - 2) x2 + y2 β 10x β 16y + 7 = 0 Centre ( - g3, - f3) = (5, 8) ...

### Find the equation of the circle concentric with the circle and of double its area.

Answer: Two or more circles are said to be concentric If they have the same centre and different radii. Given, x2 + y2 - 6x + 12y + 15 = 0 Radius r = The...

### Find the equation of the circle concentric with the circle and which touches the y-axis.

Answer: The general equation of the circle is, x2 + y2 + 2gx + 2fy + c = 0 Radius, r = $\begin{array}{l} r = \sqrt {{{(2)}^2} + {{(3)}^2} - ( - 3)} \\ r =...

### Find the equation of the circle which passes through the points (1, 3) and (2, – 1), and has its centre on the line 2x + y β 4 = 0.

Answer: The equation of a circle: x2 + y2 + 2gx + 2fy + c = 0β¦(i) Putting (1, 3) & (2, - 1) in (i) 2g + 6f + c = - 10..(ii) 4g - 2f + c = - 5..(iii) The centre lies on the given straight line, (...

### Show that the points A(1, 0), B(2, – 7), c(8, 1) and D(9, – 6) all lie on the same circle. Find the equation of this circle, its centre and radius.

Answer: The general equation of a circle: (x - h)2 + (y - k)2 = r2 β¦(i) (h, k) is the centre and r is the radius. Putting (1, 0) in (i) (1 - h)2 + (0 - k)2 = r2 h2 + k2 +...

### Find the equation of the circle concentric with the circle and passing through the point P(5, 4).

Answer: Two or more circles are said to be concentric If they have the same centre and different radii. Given, x2 + y2 + 4x + 6y + 11 = 0 The concentric circle...

### Find the equation of the circle which is circumscribed about the triangle whose vertices are A( – 2, 3), b(5, 2) and C(6, – 1). Find the centre and radius of this circle.

Answer: The general equation of a circle: (x - h)2 + (y - k)2 = r2 ...(i) (h, k) is the centre r is the radius Putting A( - 2, 3), B(5, 2) and c(6, - 1) in the equation, h2 + k2 + 4h - 6k + 13 = r2...

### Find the equation of the circle passing through the points (20, 3), (19, 8) and (2, – 9) Also, find the centre and radius.

Answer: The required circle equation, Using Laplace Expansion, 102(x2 + y2) - 1428x - 612y - 11322 = 0 x2 + y2 - 14x -6y - 11 = 0 The equation with centre = (7, 3) Radius...

### Find the equation of the circle passing through the points (i) (0, 0), (5, 0) and (3, 3) (ii) (1, 2), (3, – 4) and (5, – 6). Also, find the centre and radius

Answers: (i) The required circle equation, Using Laplace Expansion, 15(x2 + y2) - 75x - 15y = 0 x2 + y2 - 5x - y =0 The equation with centre = (2.5, 0.5) Radius = (ii) The...

### Show that the equation does not represent a circle.

Answer: Radius = The radius is negative which is not possible x2 + y2 - 3x + 3y + 10 = 0 does not represent a circle.

### Show that the equation represents a point circle. Also, find its centre.

Answer: The general equation of a circle is, x2 + y2 + 2gx + 2fy + c = 0 c, g, f are constants. x2 + y2 + 2x + 10y + 26 = 0 The equation represents a circle with 2g = 2 βg = 1, 2f = 10 βf = 5 and c...

### Show that the equation represents a circle. Find its centre and radius.

Answer: The general equation of a conic is, ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 a, b, c, f, g, h are constants For a circle, a = b and h = 0. The equation is, x2 + y2 + 2gx + 2fy + c = 0 ...

### Show that the equation represents a circle. Find its centre and radius.

Answer: The general equation of a conic is, ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 a, b, c, f, g, h are constants For a circle, a = b and h = 0. The equation is, x2 + y2 + 2gx + 2fy + c = 0 x2 + y2 +...

### Show that the equation represents a circle. Find its centre and radius.

Answer: The general equation of a conic is, ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 a, b, c, f, g, h are constants For a circle, a = b and h = 0. The equation is, x2 + y2 + 2gx + 2fy + c = 0 x2 + y2 β...