Home
Class 12
MATHS
Prove that the angle between the tangent...

Prove that the angle between the tangents from `(alpha,beta)` to the circle `x^2+y^2=a^2` is `2tan^(-1)(a/(sqrt(S_1)))` , where `S_1=alpha^2+beta^2-a^2`

A

`tan^(-1)((a)/(sqrt(S_(1))))`

B

`2tan^(-1)((a)/(sqrt(S_(1))))`

C

`2 tan^(-1)((sqrt(S_(1)))/(a))`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Let PT and PT' be the tangents drawn from `P(alpha, beta)` to the circle `x^(2)+y^(2)=a^(2)`, and let `angleTPT'=theta.` If O is the centre of the circle, then
`angleTPO=angleT'PO=theta//2`
`:. tan. (theta)/(2)=(OT)/(OP)=(a)/(sqrt(S_(1)))rArr theta = 2 tan^(-1)((a)/(sqrt(S_(1)))`
Promotional Banner

Topper's Solved these Questions

  • CIRCLES

    OBJECTIVE RD SHARMA ENGLISH|Exercise Section I - Solved Mcqs|108 Videos

  • CIRCLES

    OBJECTIVE RD SHARMA ENGLISH|Exercise Section-I (Solved MCQs)|1 Videos

  • CARTESIAN PRODUCT OF SETS AND RELATIONS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|30 Videos

  • COMPLEX NUMBERS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|58 Videos

Similar Questions

Explore conceptually related problems

If S_(1)=alpha^(2)+beta^(2)-a^(2) , then angle between the tangents from (alpha, beta) to the circle x^(2)+y^(2)=a^(2) , is

If the chord of contact of the tangents from the point (alpha, beta) to the circle x^(2)+y^(2)=r_(1)^(2) is a tangent to the circle (x-a)^(2)+(y-b)^(2)=r_(2)^(2) , then

Find the condition that the chord of contact of tangents from the point (alpha, beta) to the circle x^2 + y^2 = a^2 should subtend a right angle at the centre. Hence find the locus of (alpha, beta) .

Find the condition that the chord of contact of tangents from the point (alpha, beta) to the circle x^2 + y^2 = a^2 should subtend a right angle at the centre. Hence find the locus of (alpha, beta) .

I If a point (alpha, beta) lies on the circle x^2 +y^2=1 then the locus of the point (3alpha.+2, beta), is

(1+tan alpha tan beta)^2 + (tan alpha - tan beta)^2 =

Prove that: tan(alpha+beta)tan(alpha-beta)=(sin^2 alpha-sin^2 beta)/(cos^2 alpha-sin^2 beta)

If tan beta = 3 tan alpha, prove that tan (alpha + beta) = (2sin 2beta)/(1+cos 2beta)

If tangent of y^(2)=x at (alpha,beta) , where betagt0 is also a tangent of ellipse x^(1)2y^(2)=1 then value of alpha is

alpha and beta are acute angles and cos2alpha = (3cos2beta-1)/(3-cos2beta) then tan alpha cot beta =

OBJECTIVE RD SHARMA ENGLISH-CIRCLES-Chapter Test
  1. Prove that the angle between the tangents from (alpha,beta) to the cir...

    Text Solution

    |

  2. The two circles x^2 + y^2 -2x+6y+6=0 and x^2 + y^2 - 5x + 6y + 15 = 0 ...

    Text Solution

    |

  3. The two circles x^(2)+y^(2)-2x-2y-7=0 and 3(x^(2)+y^(2))-8x+29y=0

    Text Solution

    |

  4. The centre of a circle passing through (0,0), (1,0) and touching the C...

    Text Solution

    |

  5. The circle x^2+y^2=4 cuts the circle x^2+y^2+2x+3y-5=0 in A and B, The...

    Text Solution

    |

  6. One of the limit point of the coaxial system of circles containing x^(...

    Text Solution

    |

  7. A circle touches y-axis at (0, 2) and has an intercept of 4 units on t...

    Text Solution

    |

  8. The equation of the circle whose one diameter is PQ, where the ordinat...

    Text Solution

    |

  9. The circle x^(2)+y^(2)+4x-7y+12=0 cuts an intercept on Y-axis is of le...

    Text Solution

    |

  10. Prove that the equation of any tangent to the circle x^2+y^2-2x+4y-4=0...

    Text Solution

    |

  11. The angle between the pair of tangents from the point (1, 1/2) to the...

    Text Solution

    |

  12. The intercept on the line y=x by the circle x^(2)+y^(2)-2x=0 is AB. Eq...

    Text Solution

    |

  13. If 3x+y=0 is a tangent to a circle whose center is (2,-1) , then find ...

    Text Solution

    |

  14. The locus of the midpoint of a chord of the circle x^2+y^2=4 which sub...

    Text Solution

    |

  15. Two tangents to the circle x^(2) +y^(2) = 4 at the points A and B meet...

    Text Solution

    |

  16. A tangent is drawn to the circle 2(x^(2)+y^(2))-3x+4y=0 and it touch...

    Text Solution

    |

  17. the length of the chord of the circle x^(2)+y^(2)=25 passing through ...

    Text Solution

    |

  18. If the points A(2, 5) and B are symmetrical about the tangent to the c...

    Text Solution

    |

  19. The equation of the circle of radius 2 sqrt(2) whose centre lies on th...

    Text Solution

    |

  20. Prove that the maximum number of points with rational coordinates on a...

    Text Solution

    |

  21. The equation of a circle C is x^(2)+y^(2)-6x-8y-11=0. The number of re...

    Text Solution

    |