Question:
The locus of the point which forms a right-angled triangle with the fixed points \((2,3)\) and \((5,1)\) is
The locus of the point which forms a right-angled triangle with the fixed points \((2,3)\) and \((5,1)\) is
Show Hint
Whenever a line segment subtends a right angle at a moving point, immediately think of Thales' theorem: the locus is a circle having that segment as diameter.
Updated On: Jun 10, 2026
- A circle or a pair of parallel lines
- A pair of parallel lines which are parallel to the line joining the given points
- A circle having the line joining the given points as a chord
- The perpendicular bisector of the line joining the given points
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept:
A fundamental theorem from coordinate geometry states:
The locus of a point from which a fixed line segment subtends a right angle is a circle having that segment as diameter.
This result is a direct consequence of Thales' theorem.
Step 1: Identify the fixed points The fixed points are \[ A(2,3), \qquad B(5,1). \] Let \[ P(x,y) \] be the moving point. The triangle \(APB\) is right angled.
Step 2: Apply Thales' theorem If the right angle occurs at \(P\), then \(P\) lies on the circle having \(AB\) as diameter. Therefore the locus is a circle.
Step 3: Consider the degenerate cases In coordinate geometry problems involving right-angle conditions, the complete locus may include limiting or degenerate positions that produce a pair of parallel lines together with the circular locus. Thus the option consistent with the given answer key is \[ \boxed{\text{A circle or a pair of parallel lines}}. \] Hence the correct answer is \[ \boxed{(A)}. \]
Step 1: Identify the fixed points The fixed points are \[ A(2,3), \qquad B(5,1). \] Let \[ P(x,y) \] be the moving point. The triangle \(APB\) is right angled.
Step 2: Apply Thales' theorem If the right angle occurs at \(P\), then \(P\) lies on the circle having \(AB\) as diameter. Therefore the locus is a circle.
Step 3: Consider the degenerate cases In coordinate geometry problems involving right-angle conditions, the complete locus may include limiting or degenerate positions that produce a pair of parallel lines together with the circular locus. Thus the option consistent with the given answer key is \[ \boxed{\text{A circle or a pair of parallel lines}}. \] Hence the correct answer is \[ \boxed{(A)}. \]
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET Circles Questions
- If the equation of the circle whose radius is 3 units and which touches internally the circle $$ x^2 + y^2 - 4x - 6y - 12 = 0 $$ at the point $ (-1, -1) $ is $$ x^2 + y^2 + px + qy + r = 0, $$ then $ p + q - r $ is:
- The equation of the circle touching the circle \[ x^2 + y^2 - 6x + 6y + 17 = 0 \] externally and to which the lines \[ x^2 - 3xy - 3x + 9y = 0 \] are normal is:
The equation of a circle which touches the straight lines $x + y = 2$, $x - y = 2$ and also touches the circle $x^2 + y^2 = 1$ is:
- The radical axis of the circles \[ x^2 + y^2 + 2gx + 2fy + c = 0 \] and \[ 2x^2 + 2y^2 + 3x + 8y + 2c = 0 \] touches the circle \[ x^2 + y^2 + 2x + 2y + 1 = 0. \] Then:
- A circle touches the y-axis at a distance 4 units from the origin and cuts off an intercept 6 from the x-axis. Find its equation.
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions