Question

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

Solution

Correct option is

 

The equation of any tangent to the ellipse    

       

      

Let P(hk) be the point of intersection of tangents. 

If (i) passes through P(hk), then    

      

  

This gives two values of m, say m1 and m2. These values represent the slopes of the tangents passing through P

If the tangents drawn from P make complementary angles with x-axis, then 

       

  

.

Hence, the locus of (hk) is x2 - y2 = a2 – b2

SIMILAR QUESTIONS

Q1

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

Q2

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 

Q3

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   

Q4

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

Q5

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

Q6

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

Q7

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)

Q8

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

Q9

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

Q10

The locus of the foot of the perpendicular drawn from the centre of the ellipse  on any tangent is