Question

Two straight lines pass through the fixed points  and have gradients whose product is k > 0. Show that the locus of the points of intersection of the lines is a hyperbola.

Solution

Correct option is

 

y = m1(x – a), y = m2(x + a) where m1m2 = k, given. In order to find the locus of their point of intersection we have to eliminate the unknown m1and m2.

Multiple, we get

      y2 = m1m2(x2 – a2)       or       y2 = k(x2 – a2)

or  which represent a hyperbola.

SIMILAR QUESTIONS

Q1

Let be two points on the hyperbola . If (h, k) is thepoint of intersection of the normal’s at P and Qk is equal to

Q2

Let be two points on the hyperbola . If (h, k) is thepoint of intersection of the normal’s at P and Qk is equal to

Q3

Let two perpendicular chords of the ellipse  each passing through exactly one of the foci meet at a point P. If from P two tangents are drawn to the hyperbola , then 

Q4

If x = 9 is the chord of contact of the hyperbola x2 – y2 = 9, then the equation of the corresponding pair of triangle is.

Q5

Find the equations of tangents to the hyperbola x2 – 4y = 36 which are perpendicular to the line x – y + 4 = 0

Q6

Find the coordinates of foci, the eccentricity and latus rectum. Determine also the equation of its directrices for the hyperbola

4x2 – 9y2 =36.

Q7

Find the distance from A(4, 2) to the points in which the line 3x – 5= 2 meets the hyperbola xy = 24. Are these points on the same side of A?

Q8

The asymptotes of the hyperbola  makes an angle 600 with x-axis. Write down the equation of determiner conjugate to the diameter y = 2x.

Q9

Find the equation of the triangles drawn from the point (–2, –1) to the hyperbola 2x2 – 3y2 = 6.

Q10

 

Find the range of ‘a’ for which two perpendicular tangents can be drawn to the hyperbola from any point outside the hyperbola

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