The hypotenuse of a right-angled triangle has | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(N/A) Let the vertices of the right-angled triangle be $A(1, 3)$,$B(-4, 1)$,and $C(x, y)$,where $\angle C = 90^{\circ}$.Since $C$ lies on a circle wit

Vedclass · Accepted Answer

Vedclass · Answer

(N/A) Let the vertices of the right-angled triangle be $A(1, 3)$,$B(-4, 1)$,and $C(x, y)$,where $\angle C = 90^{\circ}$.
Since $C$ lies on a circle with diameter $AB$,the locus of $C$ is $(x - 1)(x + 4) + (y - 3)(y - 1) = 0$.
There are infinitely many such triangles depending on the choice of point $C$ on this circle.
Let the slope of leg $AC$ be $m$. Then the slope of leg $BC$ is $-\frac{1}{m}$ (since $AC \perp BC$).
The equation of line $AC$ passing through $(1, 3)$ is $y - 3 = m(x - 1)$.
The equation of line $BC$ passing through $(-4, 1)$ is $y - 1 = -\frac{1}{m}(x + 4)$.
For any real value of $m$,these two lines represent the legs of a right-angled triangle with the given hypotenuse.

The hypotenuse of a right-angled triangle has its endpoints at the points $(1, 3)$ and $(-4, 1)$. Find the equations of the legs (perpendicular sides) of the triangle.

Similar Questions

$A(a, 0)$ is a fixed point and $\theta$ is a parameter such that $0 < \theta < 2 \pi$. If $P(a \cos \theta, a \sin \theta)$ is a point on the circle $x^2+y^2=a^2$ and $Q(b \sin \theta, -b \cos \theta)$ is a point on the circle $x^2+y^2=b^2$,then the locus of the centroid of the triangle $APQ$ is

The centres of a set of circles,each of radius $3$,lie on the circle ${x^2} + {y^2} = 25$. The locus of any point in the set is

$A(4,3)$ and $B(2,5)$ are two points. If $P$ is a variable point on the same side as the origin with respect to the line $AB$ and is at most at a distance of $5$ units from the midpoint of $AB$,then the locus of $P$ is

Let $AB$ be the diameter of a semicircle $S$. The locus of the centres of circles which are tangent to $AB$ and to $S$ is an arc of

The locus of the centre of the circle,which cuts the circle $x^2+y^2-20x+4=0$ orthogonally and touches the line $x=2$,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module