Aber das ist English ! ðŸ™‚

**Question 1:**

Find the equation to that tangent to the parabola which is parallel to the straight line . Find also its point of contact.

**Question 2:**

If P, Q and R are three points on the parabola whose coordinates are in geometric progression, prove that the tangents at P and R meet on the ordinate of Q.

**Question 3:**

is a double ordinate of the parabola . Prove that the locus of the point of intersection of the normal at P and straight line through parallel to the axis is the parabola .

**Question 4:**

Show that in a parabola, the length of the focal chord varies inversely as the square of the distance of the vertex of the parabola from the focal chord.

**Question 5:**

Prove that the equation represents a parabola whose axis is parallel to the x-axis.. Find its vertex.

**Question 6:**

If the line touches the parabola , find the length of the latus rectum.

**Question 7a:**

Prove that the circle described on any focal chord of a parabola as the diameter touches the directrix of the parabola.

**Question 7b:**

Show that the locus of the point, such that two of the normals drawn from it to the parabola coincide is .

**Question 7c:**

If the normals at three points A, B and C on the parabola pass through the point and cut the axis of the parabola in P, Q and R so that OP, OQ, OR are in AP, O being the vertex **o**f the parabola, prove that the locus of the point S is .

**Question 8:**

If P, Q and R be three conormal points on the parabola , the normals at which pass through the point T, and S is the focus of the parabola, then prove that

**Question 9:**

The tangents at P and Q to the parabola meet in T and the corresponding normals meet in R. If the locus of T is a straight line parallel to the axis of the parabola, prove that the locus of R is a straight line normal to the parabola.

**Question 10a:**

A variable chord PQ of the parabola is drawn parallel to the line . If the parameters of the points P and Q on the parabola be and , then . Also, show that the locus of the point of intersection of the normals at P and Q is , which is itself a normal to the parabola.

**Question 10b:**

PQ is a chord of the parabola whose right bisector meets the axis in M and the ordinate of the mid-point of PQ meets the axis in L. Show that LM is constant and find LM.

**Question 11:**

Prove that the locus of a point P such that the slopes , , of the three normals drawn to the parabola from P be connected by the relation is .

**Question 12a:**

If the perpendicular drawn from P on the polar of P with respect to the parabola, , prove that the locus of P is the straight line .

**Question 12b:**

Prove that the locus of the poles of the tangent to the parabola w.r.t. the circle is .

**Question 13:**

Tangents are drawn to the parabola at the points P and Q whose inclination to the axis are , . If A be the vertex of the parabola and the circles on AP and AQ as diameters intersect in R and AR be inclined at an angle to the axis, then prove that .

**Question 14:**

Through the vertex O of a parabola chords OP and OQ are drawn at right angles to one another. Prove that for all positions of P, PQ cuts the axis of the parabola at a fixed point. Also, find the locus of the middle point of PQ.

**Question 15a:**

Prove that the area of a triangle inscribed in a parabola is twice the area of the triangle formed by the tangents at the vertices of the triangle.

**Question 15b:**

Normals are drawn to the parabola at points A, B, and C whose parameter are , , respectively. If these normals enclose a triangle PQR, then prove that its area is . Also, prove that .

**Question 16:**

Find the locus of the middle points of the chords of the ellipse which are drawn through the positive end of the minor axis.

**Question 17:**

Prove that the sum of the squares of the perpendiculars on any tangent to the ellipse from the points on the minor axis, each at a distance from the centre, is .

**Question 18:**

If and are two variable points on the ellipse such that (some constant), then prove that the tangent at is parallel to PQ.

**Question 19:**

P is a variable point on the ellipse whose foci are the points , and the eccentricity is e. Prove that the locus of incentre of is an ellipse whose eccentricity is .

**Question 20:**

Consider the family of circles , . If in the first quadrant, the common tangent to a circle of this family and the ellipse meets the coordinate axes at A and B, then find the equation of the locus of the mid-point of AB.

**Question 21:**

Let P be a point on the ellipse , . Let the line parallel to y-axis passing through P meet the circle at the point Q such that P and Q are on the same side of the x-axis. For two positive real numbers, r and s, find the locus of the point R on PQ such that as P varies over the ellipse.

**Question 22:**

If the eccentric angles of points P and Q on the ellipse be and and be the angle between the normals at P and Q, then prove that the eccentricity e is given by .

**Question 23:**

A series of hyperbolas are such that the length of their transverse axis is 2a. Prove that the locus of a point P on each, such that its distance from transverse axis is equal to its distance from an asymptote is the curve:

.

**Question 24:**

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

**Question 25:**

If the tangent at the point on the hyperbola cuts the auxillary circle in points whose coordinates are and , then show that q is harmonic mean of and .

**Question 26:**

Show that the locus of poles with respect to the parabola of the tangents to the hyperbola to the ellipse .

**Question 27:**

The point P on the hyperbola with focus S is such that the tangent at P, the latus rectum through S and one asymptote are concurrent. Prove that SP is parallel to other asymptote.

**Question 28:**

If a triangle is inscribed in a rectangular hyperbola, prove that the orthocentre of the triangle lies on the curve.

**Question 29:**

A series of chords of the hyperbola touch the circle on the line joining the foci as diameter. Show that the locus of the poles of these chords with respect to the hyperbola is .

**Question 30:**

Prove that the chord of the hyperbola which touches the conjugate hyperbola is bisected at the point of contact.

*Cheers,*

Nalin Pithwa.