Question

If 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.

• A. $x^2 + y^2 - 65 = 0$
• B. $x^2 + y^2 - 74 = 0$
• C. $x^2 + y^2 - 61 = 0$
• D. $x^2 + y^2 - 52 = 0$

Answer 1

The Correct Option is A

The coordinates of \( C \) are determined using the slopes of the sides and the equations of the lines.

1. Slope of \( AD \):

\[ \text{Slope of } AD = 3 \]

2. Slope of \( BC \):

\[ \text{Slope of } BC = -\frac{1}{3} \]

Equation of \( BC \):

\[ 3y + x - 17 = 0 \]

3. Slope of \( BE \):

\[ \text{Slope of } BE = 1 \]

4. Slope of \( AC \):

\[ \text{Slope of } AC = -1 \]

Equation of \( AC \):

\[ x + y - 3 = 0 \]

Solving the equations yields:

\[ \text{Point } C \text{ is } (-4, 7) \]

As \( C \) is on the circle, the equation \( x^2 + y^2 - 65 = 0 \) holds.