The Locus Of A Variable Point Whose Distance From (–2, 0) Is  times Its Distance From The Line , Is

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

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

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

4x² – 9y² =36.

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

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

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.

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

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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Find the hyperbola whose asymptotes are 2x – y = 3 and 3x + y – 7 = 0 and which passes through the point (1, 1).

A variable straight line of slope 4 intersects the hyperbola xy = 1 at two points. The locus of the point which divides the line segment between these two points in the ratio 1 : 2 is