Question

The locus of the point of intersection of tangents to the ellipse  at the points whose eccentric angles differ by  is  

Solution

Correct option is

 

 be two points on the ellipse.

The equations of tangents to the ellipse at points P and Q are   

       

and, 

         

Let P(hk) be the point of intersection of (i) and (ii). Then, 

         

  

  

Hence, the locus of (hk) is .    

 

ALITER: The coordinates of the point of intersection of tangents at points having eccentric angles α and β are

             

Thus, if (hk) is the point of intersection of tangents at points whose eccentric angles differ by , then      

       

Hence, the locus of (hk) is

       .

SIMILAR QUESTIONS

Q1

The area of the quadrilateral formed by the tangents at the end-points of latusrecta to the ellipse 

Q2

If α – β = constant, then the locus of the point of intersection of tangents at  to the ellipse   

Q3

Let S(3, 4) and S(9, 12) be two foci of an ellipse. If the coordinates of the foot of the perpendicular from focus S to a tangent to the ellipse is (1, –4), then the eccentricity of the ellipse is 

Q4

Let S and S’ be two foci of the ellipse . If a circle described on SS’ as diameter intersects the ellipse in real and distinct points, then the eccentricity e of the ellipse satisfies   

Q5

If PSQ is a focal chord of the ellipse , then the harmonic mean of SP and SQ is  

Q6

If PSQ is a focal chord of the ellipse 16x2 + 25y2 = 400 such that SP = 8, then SQ =

Q7

If S and S’ are two focii of the ellipse 16x2 + 25y2 = 400 andPSQ is a focal chord such that SP = 16, then SQ = 

Q8

Tangent at a point on the ellipse  is drawn which cuts the coordinates axes at A and B. The minimum area of the triangleOAB is (O being origin)

Q9

The locus of the foot of the perpendicular from the foci on any tangent to the ellipse 

Q10

The locus of the point of intersection of tangents to the ellipse , which make complementary angles with x-axis, is