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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).
6 years ago
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Dear student, 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 Please feel free to ask your queries here. We are all IITians and here to help you in your IIT JEE preparation. All the best. Win exciting gifts by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian. Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar respectively : Click here to download the toolbar.. Askiitians Expert Sagar Singh B.Tech, IIT Delhi
Dear student,
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
Please feel free to ask your queries here. We are all IITians and here to help you in your IIT JEE preparation.
All the best.
Win exciting gifts by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.
Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar respectively : Click here to download the toolbar..
Askiitians Expert
Sagar Singh
B.Tech, IIT Delhi
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