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JEE Mathematics 2025 Question with Solution

Let the area of a triangle △PQR\triangle PQR with vertices P(5,4)P(5, 4), Q(−2,4)Q(-2, 4), and R(a,b)R(a, b) be 3535 square units. If its orthocenter and centroid are O(2,145)O(2, \frac{14}{5}) and C(c,d)C(c, d) respectively, then c+2dc + 2d is equal to:

  • A

    73\frac{7}{3}

  • B

    33

  • C

    22

  • D

    83\frac{8}{3}

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: P(5,4)P(5,4), Q(−2,4)Q(-2,4), R(a,b)R(a,b), area of △PQR=35\triangle PQR = 35 square units, orthocenter O(2,145)O\left(2,\frac{14}{5}\right), and centroid C(c,d)C(c,d).

Find: c+2dc+2d.

Using the area formula for triangle PQRPQR:

Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\text{Area} = \frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|

So,

35=12∣5(4−b)−2(b−4)+a(4−4)∣35 = \frac{1}{2}\left|5(4-b) - 2(b-4) + a(4-4)\right| 35=12∣20−5b−2b+8∣=12∣28−7b∣=72∣4−b∣35 = \frac{1}{2}\left|20-5b-2b+8\right| = \frac{1}{2}|28-7b| = \frac{7}{2}|4-b|

Hence,

72∣4−b∣=35  ⟹  ∣4−b∣=10\frac{7}{2}|4-b| = 35 \implies |4-b| = 10

Therefore,

b=−6orb=14b=-6 \quad \text{or} \quad b=14

Using orthocenter and centroid

Since PP and QQ both have y=4y=4, the side PQPQ is horizontal. Therefore the altitude from R(a,b)R(a,b) is vertical, so its equation is

x=ax=a

The orthocenter lies on every altitude, and given orthocenter is O(2,145)O\left(2,\frac{14}{5}\right), we get

a=2a=2

Now the centroid of triangle PQRPQR is

c=5+(−2)+a3=3+a3,d=4+4+b3=8+b3c = \frac{5+(-2)+a}{3} = \frac{3+a}{3}, \qquad d = \frac{4+4+b}{3} = \frac{8+b}{3}

Substituting a=2a=2,

c=53c=\frac{5}{3}

Case 1: b=−6b=-6

d=8−63=23d=\frac{8-6}{3}=\frac{2}{3}

Thus,

c+2d=53+2(23)=5+43=93=3c+2d = \frac{5}{3}+2\left(\frac{2}{3}\right)=\frac{5+4}{3}=\frac{9}{3}=3

Case 2: b=14b=14

d=8+143=223d=\frac{8+14}{3}=\frac{22}{3}

Then,

c+2d=53+2(223)=493c+2d = \frac{5}{3}+2\left(\frac{22}{3}\right)=\frac{49}{3}

The solution states that checking the orthocenter condition gives a contradiction, so this case is invalid.

Therefore, the correct value is 33 and the correct option is B.

Common mistakes

  • Using the area formula incorrectly by mishandling signs is a common mistake. Here −2(b−4)-2(b-4) must be expanded carefully. A sign error gives wrong values of bb. Expand each term step by step before simplifying.

  • Assuming the centroid formula without substituting the correct vertex coordinates can lead to errors. The centroid is the average of the three vertices, so use c=5+(−2)+a3c=\frac{5+(-2)+a}{3} and d=4+4+b3d=\frac{4+4+b}{3} exactly.

  • Ignoring the geometry of the orthocenter is incorrect. Since PQPQ is horizontal, the altitude from RR is vertical, so the orthocenter must satisfy x=ax=a. Use this to get a=2a=2 from the orthocenter coordinate.

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