Mathematics

Mathematics

Problem

For a real number p, let A(p,2p+3) and B(5p−2,−p+7) be two points. A point D has coordinates (13p−45, 4p+235).\bigl(\frac{13p-4}{5},\,\frac{4p+23}{5}\bigr). a. (i) Prove that, for every real value of p, the point D divides the segment AB internally in the constant ratio 2:3. b. (ii) Taking p=4, write the numerical coordinates of A, B and D. c. (iii) For p=4, compute the lengths AD and DB and verify that their ratio equals 2:3.
Intuitive explanation

The key insight is recognizing that DD lies on the line segment ABAB if and only if it can be expressed as a weighted average of AA and BB's coordinates—that is, D=tA+(1−t)BD = tA + (1-t)B for some t∈[0,1]t \in [0,1]. This means the three points are collinear and DD is between them. Set up equations for both the xx and yy coordinates and solve for pp and tt to determine the configuration.

Intuitive explanation

The key insight is recognizing that DD lies on the line segment ABAB if and only if it can be expressed as a weighted average of AA and BB's coordinates—that is, D=tA+(1−t)BD = tA + (1-t)B for some t∈[0,1]t \in [0,1]. This means the three points are collinear and DD is between them. Set up equations for both the xx and yy coordinates and solve for pp and tt to determine the configuration.

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