Let $Δ$ PQR be a right angled isoceles triangle which is right angled at $P (2, 1)$ . lf the equation of the line OR is $2 x + y = 3$ , then the equation representing the pair of lines PQ and PR is

3 x^{2} - 3 y^{2} + 8 x y + 20 x + 10 y + 25 = 0

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3 x^{2} - 3 y^{2} + 8 x y - 20 x - 10 y + 25 = 0

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3 x^{2} - 3 y^{2} + 8 x y - 10 x + 15 y + 20 = 0

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3 x^{2} - 3 y^{2} + 8 x y - 10 x - 15 y + 20 = 0

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Solution

The correct option is B $3 x^{2} - 3 y^{2} + 8 x y - 20 x - 10 y + 25 = 0$
Since the triangle is isosceles, and right angled at P, therefore
$P Q = P R$
And
$P Q$ is perpendicular to $P R$ .
The the combined equation of both PQ and PR should be of the form,
$a x^{2} - a y^{2} + b x y + c x + d y + e = 0$
Since they are perpendicular and the coefficients of $x^{2}$ and $y^{2}$ will be equal and opposite.
Now the point P must satisfy the equation.
From the above options, the point P (2,1) only satisfies the equation in Option B.
Hence the correct Option is B.

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Similar questions

Let $P Q R$ be a right angled isosceles triangle right angled at $P (2, 1)$ . If the equation of the line $Q R$ is $2 x + y = 3$ , then the equation representing the pair of lines $P Q$ and $P R$ is

Q. Let

A B C

be a right angled triangle with angle at

A (2, 1)

being

90^{o}

. If the equation of pair of the line

B C

2 x + y = 3

then the joining equation of lines

A B

and

A C

3 x^{2} - 3 y^{2} + 8 x y - 20 x - 10 y + 25 = 0

Q. The equations of the diagonals of the square formed by the pairs of straight lines

3 x^{2} + 8 x y - 3 y^{2} = 0

and

3 x^{2} + 8 x y - 3 y^{2} + 2 x - 4 y - 1 = 0

are?

Q. The equation

3 x^{2} - 8 x y - 3 y^{2} = 0

and

x - 2 y = 3

represents the sides of a triangle which is

Q. Prove that the lines represented by

3 x^{2} - 8 x y - 3 y^{2} = 0

and

x + 2 y = 3

form the sides of an isosceles right angled triangle.

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Let Δ PQR be a right angled isoceles triangle which is right angled at P(2,1). lf the equation of the line OR is 2x+y=3, then the equation representing the pair of lines PQ and PR is

Let $Δ$ PQR be a right angled isoceles triangle which is right angled at $P (2, 1)$ . lf the equation of the line OR is $2 x + y = 3$ , then the equation representing the pair of lines PQ and PR is