Tangents to the Circle Time: 16:00 p.m. to 17:00 p.m.
Q.1. Attempt any four of the following. ( 3 marks each )
Time: 16:00 p.m. to 17:00 p.m.
Q.1. Attempt any four of the following. ( 3 marks each )
a) Find the condition that the line y = mx + c is a tangent to the circle x2 + y2 = a2, hence find the equation of tangent in slope form. Also find the point of contact. b) Show that the circles x2 + y2 - 4x + 10y + 20 = 0 and x2 + y2 + 8x - 6y - 24 = 0 touch each other and find the coordinates of point of contact. c) Find the equation of the tangents to the circle x2 + y2 = 10 from (4, -2). Also find their point of contact. d) Show that the straight line 4x + 3y = 26 touches the circle x2 + y2 + 4x + 6y -12 = 0 and find the equation of the parallel tangent. e) Find the acute angle between the tangents to the circle x2 + y2 = 5 from the point (-4, 2). Q.2. Attempt any four of the following. ( 2 marks each )
a) Find the condition that the line y = mx + c is a tangent to the circle x2 + y2 = a2, hence find the equation of tangent in slope form. Also find the point of contact. b) Show that the circles x2 + y2 - 4x + 10y + 20 = 0 and x2 + y2 + 8x - 6y - 24 = 0 touch each other and find the coordinates of point of contact. c) Find the equation of the tangents to the circle x2 + y2 = 10 from (4, -2). Also find their point of contact. d) Show that the straight line 4x + 3y = 26 touches the circle x2 + y2 + 4x + 6y -12 = 0 and find the equation of the parallel tangent. e) Find the acute angle between the tangents to the circle x2 + y2 = 5 from the point (-4, 2). Q.2. Attempt any four of the following. ( 2 marks each )
a) Find the equation of normal to the circle x2 + y2 - 4x - 6y - 12 = 0 at (-1, -1). b) Find the value of k, if the line x + 2y + k = 0 touches the circle x2 + y2 = 25. c) Find the equation of tangent to the circle x2 + y2 = 10 which is perpendicular to the line x + 3y = 8. d) Find the length of the tangent segment to the circle x2 + y2 + 10x - 6y - 17 = 0 drawn from the point (5, 3). e) Find the equation of tangent to the circle 2x2 + 2y2 - 6x + 8y + 12 = 0 at (1, -2).
Be-Confident
a) Find the equation of normal to the circle x2 + y2 - 4x - 6y - 12 = 0 at (-1, -1). b) Find the value of k, if the line x + 2y + k = 0 touches the circle x2 + y2 = 25. c) Find the equation of tangent to the circle x2 + y2 = 10 which is perpendicular to the line x + 3y = 8. d) Find the length of the tangent segment to the circle x2 + y2 + 10x - 6y - 17 = 0 drawn from the point (5, 3). e) Find the equation of tangent to the circle 2x2 + 2y2 - 6x + 8y + 12 = 0 at (1, -2).
Be-Confident S.M.Popade 9822772673
S.M.Popade 9822772673
Q.1. (a) Solution: The given circle is x2 + y2 = a2................(i) and the given line is y = mx + c................(ii) Suppose that line (ii) touches to circle (i) at P(x1, y1) But the equation of tangent to the circle (i) at P(x1, y1) is xx1 + yy1 = a2 i.e. y1y = -x1x + a2 .......(iii) Since equztions (ii) and (iii) represents the same line
x1 =
a 2 m a2 and y1 = c c
But the point P(x1, y1) lies on the circle x12 + y12 = a2.
4 2 2 2 a (m + 1) = a c .
c2 = a2(m2 + 1) ( a 0) This is the required condition for tangency.
m2 a 4 2 a c2 c2
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tangent to Circle board2010 17-05-2009
a) Find the condition that the line y = mx + c is a tangent to the circle x2 + y2 = a2, hence find the equation of tangent in slope form. Also find the point of contact. b) Show that the circles x2 + y2 - 4x + 10y + 20 = 0 and x2 + y2 + 8x - 6y - 24 = 0 touch each other and find the coordinates of point of contact. c) Find the equation of ...
