First slide

Definition of a circle

Question

$Let A B be the chord of contact of the point (5, - 5) with respect to the circle x^{2} + y^{2} = 5 .$
$Then the locus of the orthocenter of Δ P A B, where P is any point on the circle, is$

Moderate

A

${(x - 1)}^{2} + {(y + 1)}^{2} = 5$
B

${(x - 1)}^{2} + {(y + 1)}^{2} = \frac{5}{2}$
C

${(x + 1)}^{2} + {(y - 1)}^{2} = 5$
D

${(x + 1)}^{2} + {(y - 1)}^{2} = \frac{5}{2}$

Solution

$Equation of chord of contact (A B) will be S_{1} = 0$
$\begin{aligned} i.e, 5 x - 5 y = 5 \\ \Rightarrow x - y = 1 \end{aligned}$
$\begin{array}{l} Let A (x_{1}, x_{1} - 1), B (x_{2}, x_{2} - 1) be the points \\ Let P be (\sqrt{5} \cos θ, \sqrt{5} \sin θ) \end{array}$
$\sin c e A, B, P l i e s o n t h e c i r c l e, c e n t r e O (0, 0) i s t h e c i r c u m c e n t r e o f t h e ∆ P A B$
$and centroid is (\frac{\sqrt{5} \cos θ + x_{1} + x_{2}}{3} + \frac{\sqrt{5} \sin θ + x_{1} - 1 + x_{2} - 1}{3})$
$\begin{array}{l} \Rightarrow Ortho centre will be \\ (\sqrt{5} \cos θ + x_{1} + x_{2}, \sqrt{5} \sin θ + x_{1} + x_{2} - 2) = (h, k) \\ \Rightarrow \sqrt{5} \cos θ = h - (x_{1} + x_{2}), \sqrt{5} \sin θ = k + 2 - (x_{1} + x_{2}) \\ \sin c e (\sqrt{5} \cos θ)^{2} + (\sqrt{5} \sin θ)^{2} = 5 \end{array}$ $[i f c i r c u m c e n t r e i s t h e o r i g i n t h e n O r h o c e n t r e = 3 G = ((x_{1} + x_{2} + x_{3}), (y_{1} + y_{2} + y_{3}))]$
$\begin{array}{l} \Rightarrow {(h - (x_{1} + x_{2}))}^{2} + {(k + 2 - (x_{1} + x_{2}))}^{2} = 5 \underset{}{\dots \dots} (1) \\ (x_{1}, x_{1} - 1), (x_{2}, x_{2} - 1) lies on circle x^{2} + (x - 1)^{2} = 5 \\ x^{2} - x - 2 = 0 \\ \Rightarrow x_{1} + x_{2} = 1 \end{array}$
$\begin{array}{l} ∴ Locus will be equation (1) : (h - 1)^{2} + (k + 1)^{2} = 5 \\ \Rightarrow (x - 1)^{2} + (y + 1)^{2} = 5 \end{array}$

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