a) The middle point of a line is (2, 3) and one end of the same line is (3,4). Find the co ordinates of the other end. [3] b) Find the equation of a straight line making 9 and 15 as x intercept and…

a) Finding the Coordinates of the Other End of a Line Segment

Given:

• Midpoint $M=(2,3)$
• One endpoint $A=(3,4)$
• Let the other endpoint be $B=(x,y)$

Step 1: Use the midpoint formula The midpoint $M$ of a line segment joining two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given by: $$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

Step 2: Substitute the given values $$(2,3)=\left(\frac{3+x}{2},\frac{4+y}{2}\right)$$

Step 3: Solve for $x$ and $y$ For the $x$-coordinate: $$2=\frac{3+x}{2}⟹4=3+x⟹x=1$$

For the $y$-coordinate: $$3=\frac{4+y}{2}⟹6=4+y⟹y=2$$

Final Answer: The coordinates of the other end are $(1,2)$.

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b) Equation of a Straight Line with Given Intercepts

Given:

• $x$-intercept $=9$
• $y$-intercept $=15$

Step 1: Understand intercept form The equation of a line in intercept form is: $$\frac{x}{a}+\frac{y}{b}=1$$ where:

• $a$ is the $x$-intercept,
• $b$ is the $y$-intercept.

Step 2: Substitute the given intercepts $$\frac{x}{9}+\frac{y}{15}=1$$

Step 3: Simplify to standard form Multiply both sides by $45$ (LCM of 9 and 15) to eliminate denominators: $$45\left(\frac{x}{9}+\frac{y}{15}\right)=45\times 1$$ $$5x+3y=45$$

Step 4: Rewrite in slope-intercept form (optional) $$3y=-5x+45⟹y=-\frac{5}{3}x+15$$

Final Answer: The equation of the line is $5x+3y=45$.