**Question 1.**

If the point on , where , where the tangent is parallel to has an ordinate , then what is the value of ?

**Question 2:**

Prove that the segment of the tangent to the curve , which is contained between the coordinate axes is bisected at the point of tangency.

**Question 3:**

Find all the tangents to the curve for that are parallel to the line .

**Question 4:**

Prove that the curves , where , and , where is a differentiable function have common tangents at common points.

**Question 5:**

Find the condition that the lines may touch the curve .

**Question 6:**

Find the equation of a straight line which is tangent to one point and normal to the point on the curve , and .

**Question 7:**

Three normals are drawn from the point to the curve . Show that c must be greater than 1/2. One normal is always the x-axis. Find c for which the two other normals are perpendicular to each other.

**Question 8:**

If and are lengths of the perpendiculars from origin on the tangent and normal to the curve respectively, prove that .

**Question 9:**

Show that the curve , and is symmetrical about x-axis and has no real points for . If the tangent at the point t is inclined at an angle to OX, prove that . If the tangent at meets the curve again at Q, prove that the tangents at P and Q are at right angles.

**Question 10:**

Find the condition that the curves and intersect orthogonality and hence show that the curves and also intersect orthogonally.

More later,

Nalin Pithwa.