Question

The normal at a point P on the ellipse x2 + 4y2 = 16 meets the x-axis at Q. If M is the mid-point of the line segment PQ then the locus of M intersects the latusrectums of the given ellipse at the points    

Solution

Correct option is

 

 be a point on the given ellipse. The equation of the normal to P is  

    

                                   

The meets x-axis Q(3cos θ, 0)  

Let (hk) be the coordinates of M i.e. the mid-point of PQ.

Then, 

       

Hence, the locus of M is 

Let e be the eccentricity of the given ellipse. Then, 

     

So, the equations of the latusrecta of the given ellipse are.

 . These lines intersect the ellipse (i) at  

      

Hence, required coordinates are

      .   

SIMILAR QUESTIONS

Q1

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

Q2

, be the end points of the latusrectum of the ellipse x2 + 4y2 = 4. The equations of parabolas with latusrectum PQ are 

Q3

The locus of the point of intersection of perpendicular tangents to.

Q4

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

Q5

The tangent at a point P(θ) to the ellipse  cuts the auxiliary circle at points Q and R. If QR subtends a right angle at the centre C of the ellipse, then the eccentricity of the ellipse is

Q6

Let d1 and d2 be the lengths of the perpendiculars drawn from fociS and S’ of the ellipse  to the tangent at any point P on the ellipse. Then, SP : SP’ = 

Q7

The eccentricity of an ellipse with centre at the origin and axes along the coordinate axes, is 1/2. If one of the directrices is x = 4, then the equation of the ellipse is  

Q8

If the tangents are drawn to the ellipse x2 + 2y2 = 2, then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is

Q9

If A bar of given length moves with its extremities on two fixed straight lines at right angles, then the locus of any point on the bar describes a/an

Q10

The equation  represents an ellipse, if