Through the vertex A of a parabola the chords AP and AQ are drawn at right angles. Show that the striaght line PQ intersets the axis at a fixed point.

Through the vertex 'O' of parabola y^2=4x , chords OP and OQ are drawn at right angles to one another. Show 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.

Through the vertex O of the parabola y^2=4a x , two chords O Pa n dO Q are drawn and the circles on OP and OQ as diameters intersect at Rdot If theta_1,theta_2 , and varphi are the angles made with the axis by the tangents at P and Q on the parabola and by O R , then value of cottheta_1+cottheta_2 is (a) -2tanvarphi (b) -2tan(pi-varphi) (c) 0 (d) 2cotvarphi

Two circles intersect each other at the points G and H . A straight line is drawn through the point G which intersect two circles at the points P and Q and the straight line through the point H parallel to PQ intersects the two circles at the points R and S . Prove that PQ=RS .

From a variable point on the tangent at the vertex of a parabola y^2=4a x , a perpendicular is drawn to its chord of contact. Show that these variable perpendicular lines pass through a fixed point on the axis of the parabola.

Let's try and find the answers : (i) How many strainght lines can be drawn through one fixed point.

Find the locus of the foot of the perpendicular drawn from the origin to the straight line which passes through the fixed point (a,b).

If two normals to a parabola y^2 = 4ax intersect at right angles then the chord joining their feet pass through a fixed point whose co-ordinates are:

Debanjan drew a line segment PQ of which mid-point is R and two circles are drawn with PR adn PQ as diameter . Deberatee drew a straight line through the point which interects the first circle at the point S and the second circle at the point T. Prove that PS=ST

A line is drawn through the point ( 1, 2) to meet the coordinate axes at P and Q such that it forms a triangle OPQ, where O is the origin. If the area of the triangle OPQ is least, then the slope of the line PQ is