Home
Class 12
MATHS
Through a fixed point (h, k) secants are...

Through a fixed point (h, k) secants are drawn to the circle `x^2 +y^2 = r^2`. Then the locus of the mid-points of the secants by the circle is

Text Solution

Verified by Experts

Given, circle is `x^(2)+y^(2)=r^(2)`
Equation of chord whose mid point is given, is
` T = S_(1) rArr xx_(1)-r^(2)=x_(1)^(2)+y_(1)^(2)-r^(2)`
It also passes through (h, k)`hx_(1)+ky_(1)=x_(1)^(2)+y_(1)^(2)`
`therefore` Locus of `(x_(1), y_(1))` is

`x^(2)+y^(2)=hx + ky`
Alternate Solution
Let M be the mid-point of chord AB.
`rArr CMcot MP`
`rArr ("slope of CM")* ("slope of MP")=-1`
`rArr (y_(1))/(x_(1))*(k-y_(1))/(h-x_(1))=-1`
`rArr ky_(1)-h_(1)^(2)=-hx_(1)+x_(1)^(2)`
Hence, required locus is `x^(2)+y^(2)=hx + ky`
Promotional Banner

Topper's Solved these Questions

  • CIRCLE

    IIT JEE PREVIOUS YEAR|Exercise Topic 5 Integer Answer type Question|1 Videos

  • CIRCLE

    IIT JEE PREVIOUS YEAR|Exercise Topic 5 Fill in the blanks|7 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

Through a fixed point (h,k), secant are drawn to the circle x^(2)+y^(2)=r^(2) . Show that the locus of the midpoints of the secants by the circle is x^(2)+y^(2)=hx+ky .

Through a fixed point (h, k) secants are drawn to the circle x^(2) + y^(2) =a^(2) . The locus of the mid points of the secants intercepted by the given circle is

From (3,4) chords are drawn to the circle x^(2)+y^(2)-4x=0. The locus of the mid points of the chords is :

From the origin,chords are drawn to the circle (x-1)^(2)+y^(2)=1. The equation of the locus of the mid-points of these chords is circle with radius

From points on the circle x^2+y^2=a^2 tangents are drawn to the hyperbola x^2-y^2=a^2 . Then, the locus of mid-points of the chord of contact of tangents is:

From points on the straight line 3x-4y+12=0, tangents are drawn to the circle x^(2)+y^(2)=4. Then,the chords of contact pass through a fixed point.The slope of the chord of the circle having this fixed point as its mid- point is

If a circle Passes through point (1,2) and orthogonally cuts the circle x^(2)+y^(2)=4, Then the locus of the center is:

Tangents are drawn from points on the line x+2y=8 to the circle x^(2)+y^(2)=16. If locus of mid-point of chord of contacts of the tangents is a circles S, then radius of circle S will be