Coordinate Geometry - Intercept-advanced

Find the $x$-intercept and $y$-intercept of the line given by the equation $3x - 4y = 12$.

To find the $x$-intercept, we set $y = 0$: $3x - 4(0) = 12$ $3x = 12$ $x = \frac{12}{3} = 4$ So, the $x$-intercept is $4$, and the point is $(4, 0)$.

To find the $y$-intercept, we set $x = 0$: $3(0) - 4y = 12$ $-4y = 12$ $y = \frac{12}{-4} = -3$ So, the $y$-intercept is $-3$, and the point is $(0, -3)$.

Intercepts are found by setting the opposite coordinate to zero because the axes represent the lines $x=0$ and $y=0$.

Calculate the area of the triangle formed by the line $2x + 5y = 10$ and the coordinate axes.

Step 1: Find the intercepts. For $x$-intercept ($y=0$): $2x = 10 \Rightarrow x = 5$ For $y$-intercept ($x=0$): $5y = 10 \Rightarrow y = 2$

Step 2: Use the area formula for a right triangle. $Area = \frac{1}{2} \times \text{base} \times \text{height}$ $Area = \frac{1}{2} \times |5| \times |2|$ $Area = \frac{1}{2} \times 10 = 5$ sq units.

The intercepts $5$ and $2$ represent the lengths of the base and height of the triangle formed with the origin.

Express the linear equation $4x + 3y = 24$ in the intercept form $\frac{x}{a} + \frac{y}{b} = 1$ and identify $a$ and $b$.

Given equation: $4x + 3y = 24$ To make the right-hand side equal to $1$, divide the entire equation by $24$: $\frac{4x}{24} + \frac{3y}{24} = \frac{24}{24}$ Simplify the fractions: $\frac{x}{6} + \frac{y}{8} = 1$ Comparing this with $\frac{x}{a} + \frac{y}{b} = 1$, we get: $a = 6$ ($x$-intercept) $b = 8$ ($y$-intercept)

Dividing by the constant term converts a general linear equation into intercept form, directly revealing where the line crosses the axes.

Let the $x$-intercept and $y$-intercept both be $a$. Using the intercept form: $\frac{x}{a} + \frac{y}{a} = 1$ $x + y = a$ Since the line passes through $(3, 4)$: $3 + 4 = a \Rightarrow a = 7$ The equation is $x + y = 7$. The $x$-intercept is $7$ and the $y$-intercept is $7$. $\text{Area} = \frac{1}{2} \times |7| \times |7| = \frac{49}{2} = 24.5 \text{ sq units}$

We use the property that equal intercepts mean $a = b$ in the intercept form. Solving for the unknown using the given point allows us to find the specific intercepts and then apply the area formula for a right triangle.

The $x$-intercept $a = 6$ and $y$-intercept $b = -8$. The equation in intercept form is: $\frac{x}{6} + \frac{y}{-8} = 1$ Multiplying by $24$ to clear denominators: $4x - 3y = 24$ The midpoint $M$ of $P(6, 0)$ and $Q(0, -8)$ is: $M = \left(\frac{6+0}{2}, \frac{0-8}{2}\right) = (3, -4)$

The intercepts directly provide the values for the intercept form equation. The midpoint is calculated using the standard midpoint formula for the two points where the line crosses the axes.