If a right angled triangle is inscribed in a rectangular hyperbola, prove that the tangent at the right angle is the perpendicular upon the hypotenuse.

In a rectangular hyperbola, prove that all straight lines which subtend a right angle at a point P on the curve are parallel to the normal at P.

Show that the normal to the rectangular hyperbola xy = c² at the point t meets the curve again at the point  such that .

If P₁, P₂ and P₃ are three points on the rectangular hyperbola xy = c², whose abscissas are , then the area of the triangle P₁P₂P₃ is ..............

a)

b)

c)

d)

Find the asymptotes of the hyperbola xy = hx + ky.

a) x = h and y = k

b) x = k and y = h

c) x = ±h and y = ±k

d) None of the above

Show that the straight line  always touches the hyperbola xy = c² and that its point of contact is .

A tangent to the parabola x² = 4ay meets the hyperbola xy = c² in two points P and Q. Prove that the middle point of  lies on a parabola.

Show that the pole of any tangent to the rectangular hyperbola xy = c², with respect to the circle x² + y² = a² lies on a concentric and similarly placed rectangular hyperbola.

The vertices of a quadrilateral, taken in order, are the points (0, 0), (4, 0), (6, 7) and (0, 3); find the point of intersection of the two lines joining the middle points of opposite sides.

The base of the triangle is fixed; find the locus of the vertex when one of the base angles is double of the other.

The base BC (= 2a) of a triangle ABC is fixed; the axes being BC and perpendicular to it through its middle point, find the locus of the vertex A, when the difference of the base angles is given (=α)

The base BC (=2a) of a triangle ABC is fixed; the axes being BC and perpendicular to it through its middle point, find the locus of the vertex A, when the product of the tangents of the base angles is given (=λ)

The base BC (= 2a) of a triangle ABC is fixed; the axes being BC and perpendicular to it through its middle point, find the locus of the vertex A, when the tangent of the base angle is m times the tangent of the other

The base BC (= 2a) of a triangle ABC is fixed; the axes being BC and perpendicular to it through its middle point, find the locus of the vertex A, when m times the square of one side added to n times the square of the other side is equal to a constant quantity, .

Two fixed points A and B are taken on the axes such that OA = a and OB = b; two variable points Aʹ and Bʹ are taken on the same axes. Find the locus of the intersection of ABʹ and AʹB

when

If a straight line passes through a fixed point; find the locus of the midpoint of the position of it which is intercepted between two given straight lines.

Find the locus of a point which moves so that the difference of the square of its distance from two fixed straight lines at right angles is equal to the point’s distance from a fixed straight line.

Having given the bases and the sum of the areas of a number of triangles which a have a common vertex, show that the locus of this vertex is a straight line.

Through a given point O a straight line is drawn to cut two given straight lines in R and S; find the locus of a point P on this variable straight line which is such that

Given n straight lines and a fixed point O; through O is drawn a straight line meeting these lines in the points R₁, R₂, .  .  .  .  . ,R_n and on it is taken a point R such that  Show that the locus of R is a straight line.

Let a, b, c be three real numbers satisfying

[a, b, c] = [0, 0, 0]            (E)

If the point P (a, b, c), with reference to (E), lies on the plane 2x + y + z = 1 then the value of 7a + b + c is

a) 0

b) 12

c) 7

d) 6

Show that the locus of a point, which is such that the tangents from it to two given concentric circles are inversely as the radii, is a concentric circle, the square of whose radius is equal to the sum of the squares; of the radii of the given circle.

The polar of point P with respect to the circle x² + y² = a² touches the circle (x – α)² + (y – β)² = b²; prove that the locus of P is the curve given by the equation (αx + βy – a²)² = b² (x² + y²)