7. Let a, b, c and m be fixed real numbers. Let P be the parabola with equation y = x². (a) Show that the line L with equation y = mx + c is tangent to the parabola P if and only if m² + 4c 0. In this case determine the point of contact of L with P. = (b) Let P (a, b). Show that there are two distinct lines L₁, L2 passing through the point P which are tangent to the parabola P if and only if b < a². In this case determine the points of contact Q, R of the lines L₁, L2 respectively with the parabola P. =
7. Let a, b, c and m be fixed real numbers. Let P be the parabola with equation y = x². (a) Show that the line L with equation y = mx + c is tangent to the parabola P if and only if m² + 4c 0. In this case determine the point of contact of L with P. = (b) Let P (a, b). Show that there are two distinct lines L₁, L2 passing through the point P which are tangent to the parabola P if and only if b < a². In this case determine the points of contact Q, R of the lines L₁, L2 respectively with the parabola P. =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Answer the following question:
Transcribed Image Text:7. Let a, b, c and m be fixed real numbers. Let P be the parabola with equation y = = x².
(a) Show that the line L with equation y = mx + c is tangent to the parabola P if
and only if m² + 4c 0. In this case determine the point of contact of L with
P.
=
(b) Let P = (a, b). Show that there are two distinct lines L₁, L2 passing through
the point P which are tangent to the parabola P if and only if b < a². In this
case determine the points of contact Q, R of the lines L₁, L2 respectively with the
parabola P.
(c) Show that the lines L₁, L2 are perpendicular if and only if b = -1/4
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