Question

The values of $h$ for which the equation $3 x^{2}+2 h x y-3 y^{2}-40 x$ $+30 y-75=0$ represents a pair of straight lines, are
1) 4, 4
2) 4, 6
3) 4, -4
4) 0, 4

Answer 1

Hint: An object in geometry known as a line is simply one that extends on all sides and has zero width. Simply put, a straight line is a line devoid of bends. A straight line is one that does not curve and reaches infinity on both sides.
All points on the curve are satisfied by the relationship between x and y.
The best approach to determine without using any geometrical tools if the lines are parallel, perpendicular, or at any angle is to measure the slope. The definition of a slope, the slope formula for parallel and perpendicular lines, and the slope for collinearity will all be covered in this article.

Formula Used:
The following is the general equation for a straight line:
Straight line equation: $ax + by + c = 0$
where a, b, and c are constants and x, and y are variables.

Complete step by step Solution:
Given $3 x^{2}+2 h x y-3 y^{2}-40 x+30 y-75=0$
In comparison to the general equation,
 $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$,
we obtain $a=3, h=h, b=-3, g=-20, f=15, c=-75$
It is necessary for straight lines to
$a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0$
Substitute the values
$675-600 h-675+1200+75 h^{2}=0$
Simplify the expression
$75 h^{2}-600 h+1200=0$
$=>h^{2}-8 h+16=0$
$=>(h-4)(h-4)=0$
$=>h=4, h=4$

Hence, the correct option is 1.

Note:A line's slope often indicates the steepness and direction of the line. By calculating the difference between the coordinates of the two points, $(x_1,y_1)$ and $(x_2,y_2)$, it is simple to calculate the slope of a straight line between them. The letter "m" is frequently used to denote slope.
A line's slope is defined as the ratio of the change in y coordinate to the change in x coordinate.
Both the net change in the y-coordinate and the net change in the x-coordinate are denoted by y and x, respectively.