Co-ordinate Geometry Problems: Parabola Solutions
Explore solutions to problems related to the parabola, including finding equations, proving tangent lines, and determining lengths within the context of coordinate geometry. Detailed explanations with step-by-step solutions.
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2 D Co-ordinate Geometry Lecture-12 Parabola Dated:-08.05.2020 PPT-04 UG (B.Sc., Part-1) Dr. Md. Ataur Rahman Guest Faculty Department of Mathematics M.L. Arya, College, Kasba PURNEA UNIVERSITY, PURNIA
Problems based on tangent to parabola 1. Find equation of the parabola whose focus is (5,3) and the directrix is 3x-4y+1=0. 2. Find equation of the parabola whose focus is (-6,-6) and the vertex is (-2,2). 3. Prove that the straight line touches the parabola 4. Show that the line touches the parabola and that the point of contact is 5. If the line touches the parabola , find the length of the latus-rectum. = + y a + mx c = 4 ( a x a if c + = 2 ) . y ma m + = xCos ySin p = a + = 2 2 4 0 y ax if pCos 2 ( tan a aSin , 2 tan ). = 2 4 y ax + = 2 3 1 x y
Solution of (1) Given focus and the eq. of the given directrix is Let P(x,y) be any point on the parabola. Then by the definition of parabola PF=the distance of the directrix from P(x,y) ( ) 2 2 3 4 1 5 3 3 4 + + + = ( ) y F 3 5,3 4 1 0....(1) + = x + x y ( ) ( ) + = x y 2 2 ( ) 2 3 4 25 6 1 x y ( ) ( ) 2 2 5 3 x y x + + + + = + 1 24 + + 2 2 2 2 25 ( 10 25) ( 9) 9 16 8 6 x x y y x y xy y x + + = 2 2 16 9 256 142 24 849 0 . y x y xy required eq of the parabola
Solution of (4) Solution:-The eqs. of the given st. line and the parabola are xCos ySin ySin Cos y Sin + = p = + ( ) xCos p p n = + ... (1 . ) x Si Cos Sin ax p = = Here m and c Sin = 2 4 ..... (2) and y If the line (1) touches the parabola (2) if a m pCos p aSin Cos = = = c Sin + 2 0 aSin proved
Continue If is the point of contact, then a x m ( , ) P x y 1 1 a = = = 2 tan a 1 2 2 Cos Sin 2 m 2 a a = = = 2 tan a and y 1 Cos Sin Hence the co-ordinates of the point of contact are ( tan a ) , 2 tan a 2
Solution of (5) Solution:-The eqs. of the given st. line and the parabola are 2 1..... 3 3 2 3 4 . . .. 2 . and y ax = + = 2 3 1 0 x y = + (1) y x 1 3 = = Here m and c 2 ( ) According to the question, the line (1) touches the parabola (2) a c m 3 a a = = 2 3 2 1 3 3 1 3 a = = as c 2 2 3 2 9 = = 3 a a 2 9 8 Length of latus-rectum = = = 4 4 a 9
