Question

Let $P$ and $Q$ be constants. The graphs of the lines $x + 5y = 7$ and $15x + Py = Q$ are perpendicular and intersect at the point $(-8,3).$ Enter the ordered pair $(P,Q).$

Answer 1

Answer:

(-3, -129)

Step-by-step explanation:

Given

Two lines:

$x + 5y = 7$

$15x + Py = Q$

are perpendicular to each other and intersect at point (-8,3).

To find: (P, Q)

Solution:

The two lines intersect at (-8,3).

It means, the equation of line will be satisfied when we put value of x = -8 and y = 3

Putting in the second equation, we will get an equation in P and Q:

$15\times (-8) + P \times 3 = Q\\\Rightarrow 3P = Q +120 ..... (1)$

Given that two lines are perpendicular.

It means the product of their slopes will be equal to -1.

i.e. $m_1\times m_2=-1$

Slope of a line of the form $ax+by+c=0$ is given as:

$m=-\dfrac{a}{b}$

So, slopes of given lines are:

$m_1=-\dfrac{1}{5}\\m_2=-\dfrac{15}{P}$

Using the condition:

$-\dfrac{1}{5}\times \dfrac{-15}{P}=-1\\\Rightarrow P = -3$

Putting the value of P in equation (1):

$\Rightarrow 3\times (-3) = Q +120 \\\Rightarrow Q = -9-120 = -129$

So, answer is (-3, -129)