Home
Class 12
MATHS
Consider the family ol circles x^2+y^2=...

Consider the family ol circles `x^2+y^2=r^2, 2 < r < 5` . If in the first quadrant, the common tangnet to a circle of this family and the ellipse `4x^2 +25y^2=100` meets the co-ordinate axes at A and B, then find the equation of the locus of the mid-point of AB.

A

`4x^(2)+25y^(2)=4x^(2)y^(2)`

B

`4x^(2)+25y^(2)=4`

C

`4x^(2)+25y^(2)=9x^(2)y^(2)`

D

`4x^(2)+25y^(2)=9`

Text Solution

Verified by Experts

The correct Answer is:
A

Equation of any tangent to circle `x^(2)+y^(2)=r^(2)` is
`xcostheta + y sintheta = r" "...(i)`
Suppose Eq. (i) is tangent to `4x^(2) + 25y^(2)=100`
or when `(x^(2))/(25)+(y^(2))/(4)=1 " at"(x_(1),y_(2))`
Then, Eq, (i) and `("xx"_(1))/(25)+(yy_(1))/(4)=1` are identical
`therefore (x_(1)//25)/(costheta)=((y_(1))/(4))/(sintheta)=(1)/(r)`
`rArr x_(1)(25costheta)/(r),y_(1)=(4sintheta)/(r)`
The line (i) meet the coordinates axes in A `(rsectheta, 0) and beta(0,r cosec theta)`. LEt (h,k) be mid-point of AB.
Then , " " `h=(rsectheta)/(2)and k=(rcosectheta)/(2)`
Therefore, `2h=(r)/(costheta)and 2k=(r)/(sintheta)`
`therefore x_(1)=(25)/(2h)and y_(1)=(4)/(2k)`
As `(x_(1), y_(1)) " lies on the ellipse " (x^(2))/(25)+(y^(2))/(4)=1` we get
`(1)/(25)((625)/(4h^(2)))+(1)/(4)((4)/(k^(2)))=1`
`rArr (25)/(4h^(2))+(1)/(h^(2))=1`
or `25k^(2)+4h^(2)=4h^(2)k^(2)`
Therefore, required locus is `4x^(2)+25y^(2)=4x^(2)y^(2)`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • CIRCLE

    IIT JEE PREVIOUS YEAR|Exercise Topic 5 (objective Questions I)|6 Videos
  • CIRCLE

    IIT JEE PREVIOUS YEAR|Exercise Topic 5 (objective Questions II)|1 Videos
  • CIRCLE

    IIT JEE PREVIOUS YEAR|Exercise Topic 4 Fill in the blanks|4 Videos
  • BINOMIAL THEOREM

    IIT JEE PREVIOUS YEAR|Exercise Topic 2 Properties of Binomial Coefficent Objective Questions I (Only one correct option) (Analytical & Descriptive Questions )|8 Videos
  • COMPLEX NUMBERS

    IIT JEE PREVIOUS YEAR|Exercise TOPIC 5 DE-MOIVRES THEOREM,CUBE ROOTS AND nth ROOTS OF UNITY (INTEGER ANSWER TYPE QUESTION)|1 Videos

Similar Questions

Explore conceptually related problems

Consider the family of circles x^2 + y^2= r^2 2 lt r lt 5 . If in the first quadrant, the common tangent to a circle of this family and the ellipse frac{x^(2)}{25} +frac{y^(2)}{4} =1 meets the axes at A and B then find the equation of the locus of middle point of AB.

Consider the family of circles x^(2)+y^(2)-2x-2 lambda y-8=0 passing through two fixed points AandB .Then the distance between the points AandB is

Knowledge Check

  • Consider the family of circles x^(2)+y^(2)-2x-2ay-8=0 passing through two fixed points A and B . Also, S=0 is a cricle of this family, the tangent to which at A and B intersect on the line x+2y+5=0 . The distance between the points A and B , is

    A
    4
    B
    `4 sqrt(2)`
    C
    6
    D
    8
  • Consider the family of circles x^(2)+y^(2)-2x-2ay-8=0 passing through two fixed points A and B . Also, S=0 is a cricle of this family, the tangent to which at A and B intersect on the line x+2y+5=0 . If the circle x^(2)+y^(2)-10x+2y=c=0 is orthogonal to S=0 , then the value of c is

    A
    8
    B
    9
    C
    10
    D
    12
  • Consider the family of circles : x^(2)+y^(2)-3x-4y-c_(1)=0, c_(1)inN (i=1,2,3,…,n) Also, let all circles intersects X-axis at integral points only and c_(1)ltc_(2)ltc_(3)ltc_(4)…ltc_(n)A point (x, y) is said to be integral point, if both coordinates x and y are integers. If circle x^(2)+y^(2)-3x-4y-(c_(2)-c_(1))=0 and circle x^(2)+y^(2)=r^(2) have only one common tangent, then

    A
    `r=1//2`
    B
    tangent passes through `(10,0)`
    C
    (3, 4) lies outside the circle `x^(2)+y^(2)=r^(2)`
    D
    `c_(2)=2r+c_(1)`
  • Similar Questions

    Explore conceptually related problems

    Consider the two circles C: x^2+y^2=r_1^2 and C_2 : x^2 + y^2 =r_2^2(r_2 lt r_1) let A be a fixed point on the circle C_1,say A(r_1, 0) and 'B' be a variable point on the circle C_2. Theline BA meets the circle C_2 again at C. Then The maximum value of BC^2 is

    Find the orthogonal of the family of circles x^(2) + y^(2) = 2ax each of which touches the y-axis at origin.

    Consider the equation of circle x^(2)+y^(2)-2x-2lambday -8=0 where lambda is variable then answer the following: Find the equation of a circle of this family tangents to which at these fixed points A and B of part (a) meet on the line x +2y +5=0

    Consider the family of all circles whose centers lie on the straight line y=x . If this family of circles is represented by the differential equation P y^(primeprime)+Q y^(prime)+1=0, where P ,Q are functions of x , y and y^(prime)(h e r ey^(prime)=(dy)/(dx),y^=(d^2y)/(dx^2)), then which of the following statements is (are) true? (a)P=y+x (b)P=y-x (c)P+Q=1-x+y+y^(prime)+(y^(prime))^2 (d)P-Q=x+y-y^(prime)-(y^(prime))^2

    Find the differential equation of the family of circles (x-a)^(2)+(y-b)^(2)=r^(2) , where 'a' and 'b' are arbitrary constants.