The expression p(q-r) x^2+q(r-p) xy+r(p-q) y^2 is a perfect square, where p, q, r are non zero r

The expression p(q-r) x^2+q(r-p) xy+r(p-q) y^2 is a perfect...

The expression $\mathrm{p}(\mathrm{q}-\mathrm{r}) \mathrm{x}^{2}+\mathrm{q}(\mathrm{r}-\mathrm{p}) \mathrm{xy}+\mathrm{r}(\mathrm{p}-\mathrm{q}) \mathrm{y}^{2}$ is a perfect square, where $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are non zero reals (a) Minimum value of $\mathrm{p}+\mathrm{r}$ is $2 \mathrm{q}$ (b) AM of p, r is q (c) If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are the exradii of $\triangle \mathrm{ABC}$ whose smallest side is 4 and largest side is 8 then the third side is 6 (d) The area of $\triangle \mathrm{ABC}$ described in (c) is $3 \sqrt{15}$

The expression $\mathrm{p}(\mathrm{q}-\mathrm{r}) \mathrm{x}^{2}+\mathrm{q}(\mathrm{r}-\mathrm{p}) \mathrm{xy}+\mathrm{r}(\mathrm{p}-\mathrm{q}) \mathrm{y}^{2}$ is a perfect square, where $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are non zero reals
(a) Minimum value of $\mathrm{p}+\mathrm{r}$ is $2 \mathrm{q}$
(b) AM of p, r is q
(c) If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are the exradii of $\triangle \mathrm{ABC}$ whose smallest side is 4 and largest side is 8 then the third side is 6
(d) The area of $\triangle \mathrm{ABC}$ described in (c) is $3 \sqrt{15}$

Key Concepts

Perfect Square Condition

A quadratic expression in two variables is a perfect square if it can be written as the square of a linear form. In algebra, this requires that the quadratic’s discriminant vanish, ensuring that there is no mixed remainder term when expressed in a squared form.

Discriminant of Quadratic Forms

For a quadratic expression of the form Ax² + Bxy + Cy² to be a perfect square, the discriminant B² - 4AC must equal zero. This condition assures that the expression factors neatly into the square of a binomial, which is essential when analyzing when a polynomial is a perfect square.

Arithmetic Mean

The arithmetic mean of two numbers is defined as the sum of the numbers divided by two. In the context of the problem, the condition p + r = 2q directly implies that q is the arithmetic mean of p and r, a concept often used to deduce symmetry or balance between quantities.

Exradius of a Triangle

In triangle geometry, an exradius is the radius of an excircle, which is a circle tangent to one side of the triangle and the extensions of the other two sides. The exradius is related to the triangle's area and semiperimeter, and these relationships are useful in connecting algebraic expressions to geometric properties.

Heron's Formula

Heron's formula provides a way to calculate the area of a triangle when its three side lengths are known. It states that the area is the square root of s(s-a)(s-b)(s-c), where s is the semiperimeter and a, b, and c are the side lengths. This formula is key when determining the area from side lengths that are deduced from algebraic conditions.

Transcript

00:02 The portion is set it as the expression is a perfect square where pqr are non -zero real.

00:07 We check the given options here.

00:09 If we just work on this, we have, we're supposed we take x square plus 2xy plus y square.

00:16 I take x square plus 2xy plus y plus y square.

00:20 And also if i take 4x square plus 4xy plus y square.

00:28 I take that.

00:30 Now it is 2 here 2 square.

00:32 So two square equals 4 times percent of x square times percent of y square.

00:38 This we can do.

00:40 Even if we just take here, we do four square equals four times three percent of x squared and percent of y scale.

00:47 Similarly, for the given expression, is a perfect square, then we will have q square, r negative p, all square equals 4 p r, p negative q times q negative r.

01:00 You can expand it and divide it by p square, q squared, r square.

01:06 You will get 1 over p square plus 1 over r square plus 2 over p r squared equals 4 over q 1 over p plus 1 over r negative 4 over 2.

01:18 You expand it and divide by p square q square after that step you will get it as 1 over p plus 1 over r whole square negative 4 over q 1 over p 4 over 2.

01:30 1 over p 4.

01:31 Positive 1 over r, positive 4 over cube square is equal to 0.

01:37 Or you will get it as 1 over p, positive 1 over r, negative 2 over q, 4 square is equal to 0.

01:49 This we get from the expression...

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