Tangent and Normal to a Circle





Introduction to circle

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Equation of circle

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Tangent to a circle

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Condition to be a Tangent to a circle

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Equation of tangent line to the circle

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Equation of normal line to the circle

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Length of the tangent from external point

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Exercise 1

  1. Find the equation of tangent and normal to given circle at given point
    1. x2 + y2 = 25 at (3, -4)

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    2. x2+y2=4 at (1,3)

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    3. x²+y²=4 at (2cosθ,2sinθ)

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    4. x2+y2=8 at (2,2)

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    5. x2+y2=36 at (6,0)

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    6. x²+y²+2x+4y20=0 at (3,1)

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    7. x²+y²6x8y4=0 at (8,6)

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    8. x²+y²3x+10y15=0 at (4,-11)

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    9. x2+y28x2y+12=0 at (x, -1)

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  2. Find the point where the tangent to the circle x2+y2=225 at (9, 12) crosses the x-axis. Ans: (25,0)

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  3. Find the point where the tangent to the circle x2+y2=25 at (2, 4) crosses the x-axis. Ans: (10,0)

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  4. Find the following
    1. Find the equation of tangent and normal to x²+y²=40 at the points whose (i) absciassa is 2 (ii) ordinate is -6
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    3. Find the equation of tangent to the circle 2x²+2y²=9 which makes angle 45 degree with x-axis

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    4. Find the equation of normal to the circle 2x²+2y²=9 which makes angle 45 degree with x-axis

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  5. Find the following equation of tangent to the circle
    1. x2+y2=4 which are parallel to 3x+4y5=0

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    2. x2+y2=5 which are parallel to x+2y=0

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    3. x2+y26x+4y=12 which are parallel to 4x+3y+5=0

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    4. x2+y22x4y4=0 which are parallel to 3x4y=1

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  6. Show that
    1. tangent to the circle x2+y2=100 at the points (6,8) and (8,-6) are perpendicular to each other

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    2. tangent to the circle x2+y2+4x+8y+2=0 at the points (1,-1) and (-5,-7) are parallel to each other

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  7. Find the equation of the circle whose center is (h,k) and which passes through the origin and prove that the equation of the tangent at the origin is hx+ky=0

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Exercise 2

  1. Show that the line 3x4y=25 and the circle x2+y2=25 intersect at two coincide points

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  2. Prove that the line 5x+12y+78=0 is tangent to the circle x2+y2=36

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  3. Prove that the tangent to the circle x2+y2=5 at the point (1,-2) also touches the circle x2+y28x+6y+20=0 and find the point of contact.

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  4. Prove that the line y=x+a2 touches the circle x2+y2=a2 and find the point of contact.

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  5. Find the value of k so that
    1. the line 4x+3y+k=0 may touches the circle x2+y24x+10y+4=0

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    2. the line 2xy+4k=0 touches the circle x2+y22x2y3=0

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  6. Find the condition that
    1. the line px+qy=r is tangent to the circle x2+y2=a2

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    2. the line lx+my+n=0 is tangent to the circle x2+y2+2gx+2fy+c=0

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    3. the line lx+my+n=0 is normal to the circle x2+y2+2gx+2fy+c=0

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    4. the circle x2+y2+2gx+2fy+c=0 touches the (i) x-axis (ii) y-axis

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  7. If the line lx+my=1 touches the circle x2+y2=a2, prove that the point (l,m) lies on a circle whose radius is 1a

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  8. Do the following
    1. Find the condition for the two circles x2+y2=a2 and (xc)2+y2=b2 to touch (i) externally and (ii) internally

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    2. Prove that the two circles x2+y2+2ax+c2=0 and x2+y2+2by+c2=0 touch if 1a2+1b2=1c2

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Exercise 3

  1. Find the equation to the pair of tangents drawn from the origin to the circle x2+y24x4y+7=0

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  2. Find the equation of tangents drawn from the point (11,3) to the circle x2+y2=65. Also find the angles between the two tangents.

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  3. Find the equation of tangents drawn from the origin to the circle x2+y2+10x+10y+40=0

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Exercise 4

  1. Determine the length of tangents to the circles
    1. x2+y2=25 from (3,5)

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    2. x2+y2+4x+6y19=0 from (6,4)

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  2. Determine the value of k so that the length of the tangent from (5,4) to the circle x2+y2+2ky=0 is 5.

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  3. Show that the length of the tangent drawn from any point on the circle x2+y2+2gx+2fy+c=0 to the circle x2+y2+2gx+2fy+c1=0 is c1c

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