Mathematics · Straight lines

JEE Main 2024 — 6 April, Shift 2 — Question 15

If $\mathrm{P}(6,1)$ be the orthocentre of the triangle whose vertices are $A(5,-2), B(8,3)$ and $C(h, k)$ , then the point C lies on the circle.

Answer: A

Slope of $\mathrm{AD}=3$

Slope of $B C=-\frac{1}{3}$

equation of $\mathrm{BC}=3 \mathrm{y}+\mathrm{x}-17=0$

slope of $\mathrm{BE}=1$

Slope of $\mathrm{AC}=-1$

equation of AC is $\mathrm{x}+\mathrm{y}-3=0$

The coordinates of point $C$ are $(-4,7)$ .

(-4)^2 + 7^2 = 16 + 49 = 65

Hence, the locus of point $C$ is the circle

\boxed{x^{2}+y^{2}-65=0}

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 6 April, Shift 2
Subject: Mathematics
Chapter: Straight lines
Topic: Angle between lines, perpendicular distance & distance between parallel lines, foot, image