Find the locus of the circumcenter of a triangle whose two sides are along the coordinate axes and the third side passes through the point of intersec

Question

Find the locus of the circumcenter of a triangle whose two sides are along the coordinate axes and the third side passes through the point of intersec

Find the locus of the circumcenter of a triangle whose two sides are along the coordinate axes and the third side passes through the point of intersection of the line `a x+b y+c=0` and `l x+m y+n=0.`

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

Related Questions

A variable plane passes through a fixed point `(alpha,beta,gamma)` and meets the axes at `A ,B ,a n dCdot` show that the locus of the point of interse

2 Answers 5 Views

A line with positive direction cosines passes through the point P(2, – 1, 2) and makes equal angles with the coordinate axes. The line meets the plane

2 Answers 6 Views

Find the equation of the straight line that (i)makes equal intercepts on the axes and passes through the point (2;3) (ii) passes through the point (-5

2 Answers 6 Views

Let points A,B and C lie on lines y-x=0, 2x-y=0 and y-3x=0, respectively. Also, AB passes through fixed point P(1,0) and BC passes through fixed point

2 Answers 11 Views

A straight line l with negative slope passes through (8,2) and cuts the coordinate axes at P and Q. Find absolute minimum value of 'OP+OQ`` where O is

2 Answers 9 Views

The longest side of a triangle is three times the shortest side and the third side is 2 cm shorter than the longest side if the perimeter of the trian

2 Answers 4 Views

The distance between the circumcenter and the orthocentre of the triangle whose vertices are `(0,0),(6,8),` and `(-4,3)` is `Ldot` Then the value of `

2 Answers 5 Views

In triangle ABC, the line joining the circumcenter and incenter is parallel to side AC, then `cos A + cos C` is equal to

2 Answers 6 Views

In triangle ABC, line joining the circumcenter and orthocenter is parallel to side AC, then the value of tan A tan C is equal to

2 Answers 4 Views

Two dipole of moment `5 xx 10^(-12)` C m form a cross with their axes (`- "to" +`) along the coordinate axes . Calculate the potential at a point 20 c

2 Answers 4 Views

Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Let the equation of the third line be
`(ax+by+c)+lambda(lx+my+n)=0`
`"where" lambda " is a parameter. It meets the x-axis at " (-(lambdan+c)/(lambdal+a),0) " and the "`
`"y-axis at B" (0,-(lambdan+c)/(lambdam+b))`
The triangle OAB is a right-angled triangle. Its circumcenter is the midpoint of the hypotenuse. Let it be `(alpha, beta)`. Then,
`2alpha = -((lambdan+c)/(lambdal+a))`
`"and " 2beta = -((lambdan+c)/(lambdam+b))`
`Hence ,lambda = -(2alphaa+c)/(2alphal+n) =-(2betab+c)/(2betam+n)`
Hence, the locus of `(alpha, beta)` is
`(c+2ax)/(n+2lx) =(c+2by)/(n+2my)`
or 2xy(bl-ma)=x(an-lc)+y(mc-bn)