If a leaf of a book is folded so that one corner moves along an opposite side, then prove that the line of crease will always touch parabola.

The lower corner of a leaf in a book is folded over so as to reach the inner edge of the page. Show that the fraction of the width folded over when the area of the folded part is minimum is:

One side of a rectangular piece of paper is 6 cm, the adjacent sides being longer than 6 cms. One corner of the paper is folded so that it sets on the opposite longer side. If the length of the crease is l cms and it makes an angle theta with the long side as shown, then l is

From the point where any normal to the parabola y^2 = 4ax meets the axis, is drawn a line perpendicular to the normal. Prove that the line always touches an equal parabola.

A cubical room has dimensions 4 ft xx 4 ft xx 4 ft . An insect starts from one corner O and reaches a corner on the opposite side of the body diagonal. Suppose that insect does not fly but crawls. Find the minimum distance travelled by insect to reach the destination.

The normals at P, Q, R on the parabola y^2 = 4ax meet in a point on the line y = c. Prove that the sides of the triangle PQR touch the parabola x^2 = 2cy.

One angle of a triangle is equal to one angle of another triangle and the bisectors of these two equal angles divide the opposite sides in the same ratio, prove that the triangles are similar.

One vertex of a triangle of given species is fixed and another moves along circumference of a fixed circle. Prove that the locus of the remaining vertex is a circle and find its radius.

Two circle of radii a and b touch the axis of y on the opposite side at the origin, the former being on the possitive side. Prove that the other two common tangents are given by (b-a) x+-2sqrt(ab) y-2ab=0 .

From the point where any normal to the parabola y^2 = 4ax meets the axis, is drawn a line perpendicular to this normal, prove that this normal always touches an equal parabola y^2 +4a(x-2a)=0 .

If a triangle is inscribed in an ellipse and two of its sides are parallel to the given straight lines, then prove that the third side touches the fixed ellipse.