# Q. 1 Show that the normals at the points (4a, 4a) & at the upper end of the latus ractum of the parabola y2 = 4ax intersect on the same parabola. Q. 2 Prove that the locus of the middle point of portion of a normal to y2 = 4ax intercepted between the curve & the axis is another parabola. Find the vertex & the latus rectum of the second parabola. Q.3 Find the equations Qf the tangents to the parabola y2 = 16x, which are parallel & perpendicular respectively t to the line 2x-y + 5 = 0. Find also the coordinates of their points of contact, n ' 1 i ^ / Q.4 A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola y2 = 4ax. Prove that the common chord of the circle and parabola bisects the distance between the vertex and the focus. Q. 5 Find the equations of the tangents of the parabola y2 = 12x, which passes through the point (2,5). Q.6 Through the vertex O of a parabola y2=4x, chords OP & 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. Q. 7 Let S is the focus of the parabola y2 = 4ax and X the foot of the directrix, PP' is a double ordinate of the curve and PX meets the curve again in Q. Prove that P'Q passes through focus. Q.8 Three normals to y2 = 4x pass through the point (15,12). Show that one of the normals is given by y = x - 3 & find the equations of the others. Q. 9 Find the equations of the chords of the parabola y2 = 4ax which pass through the point (-6a, 0) and which subtends an angle of 45° at the vertex. Q.IO Through the vertex O of the parabola y2=4ax, a perpendicular is drawn to any tangent meeting it at P & the parabola at Q. Show that OP • OQ = constant. Q.ll 'O' is the vertex of the parabola y2 = 4ax & L is the upper end of the latus rectum. If LH is drawn perpendicular to OL meeting OX in H, prove that the length of the double ordinate through H is 4a • Q.12 The normal at a point P to the parabola y2 = 4ax meets its axis at G. Q is another point on the parabola such that QG is perpendicular to the axis of the parabola. Prove that QG2 - PG2 = constant. Q.13 If the normal at P( 18,12) to the parabola y2= 8x cuts it again at Q, show that 9PQ = 80 VlO Q.14 Prove that, the normal to y2 = 12x at (3' 6) meets the parabola again in (27,-18) & circle on this normal chord as diameter is x2 + y2 - 30x + 12y - 27 = 0. Q.15 Find the equation of the circle which passes through the focus of the parabola x2 = 4y & touches it at the point (6,9).

## 1 Answers

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Please post one question at a time...

The tangent coordinate = (x1, 0) and (0, y1)

slope of tangent = y1/-x1 = -y1/x1

The normal coordinates = (x2, 0) and (0, y2)

slope of normal = y2 /-x2 = -y2/x2

since tangent and normal or perpendicular, the product of their slopes = -1

-(y1/x1)*(-y2/x2) = -1

y1y2 = -x1x2

x1x2 + y1y2 = 0

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