Question

Consider the family of circle x2 + y2 = r2, 2 < r < 5. If in the first quadrant the common tangent to a circle of the family and the ellipse 4x2 + 25y2 = 100 meets the coordinate axes at A and B, then find the equation of the locus of the mid point of AB.

Solution

Correct option is

 

The equation of any tangent to the ellipse

        

        

It meets the coordinate of axes at 

         

Let P(hk) be the mid-point of AB. Then  

          

  

Since (i) touches the circle  

       

Therefore, 

       

  

  

Hence the locus of P(hk) is  

        

SIMILAR QUESTIONS

Q1

Find the equations of the tangents to the ellipse  which are perpendicular to the line y + 2x = 4.

Q2

Find the locus of the foot of the perpendicular drawn from centre upon any tangent to the ellipse .

Q3

Find the locus of the points of the intersection of tangents to ellipse  which make an angle θ.

Q4

Find the locus of the poles of tangents to  with respect to the concentric ellipse .  

Q5

 

Determine the equation of major and minor axes of the ellipse  

       

Also, find its centre, length of the latusrectum and eccentricity.

Q6

 

Find the locus of the centroid of an equilateral triangle inscribed in the ellipse     

                  

Q7

If SY and S1Y1 be perpendiculars from the foci upon the tangent atP of an ellipse, then Y and Y1 lie on the auxiliary circle andSY.S1Y1 =    

Q8

Find the condition on a and b for which two distinct chords of the ellipse  passing through (a, –b) are bisected by the line x + y = b

Q9

Let P be a point on the ellipse , 0 < b < a. Let the line parallel to y-axis passes through P meet the circle  at the point Q, such that P and Q are the same side of x-axis. For two positive  real numbers r and s, find the locus of the point Ron PQ such that PR : RQ = r : s as P varies over the ellipse.  

Q10

The orbit of earth is an ellipse with eccentricity 1/60 with the sun at one focus the major axis being approximately 186 × 106 miles in length. Find the shortest and longest distance of the earth from the sun.