Mock Test 2 — Straight Lines

A straight line passes through the fixed coordinate point $P(2, 1)$. If the line cuts the positive x and y axes at points $A$ and $B$ respectively, find the minimum possible area of the triangle $OAB$ formed with the origin.

Find the equation of the line belonging to the family $(2x+3y-5) + \lambda(x-y-1) = 0$ that stands at the maximum possible perpendicular distance from the point $(3, 4)$.

The general second-degree equation $3x^2 + 8xy + 2y^2 + 4x + 2y + c = 0$ represents a pair of straight lines. Find the coordinates of their point of intersection.

A variable line drawn through the origin cuts the parallel lines $x + 2y = 3$ and $x + 2y = 6$ at points $P$ and $Q$ respectively. The ratio of the lengths $OP : OQ$ remains:

Find the reflection image of the straight line $x - 2y + 4 = 0$ across the horizontal line mirror $y = x$.

Find the joint equation of the pair of lines passing through the origin that are mutually perpendicular to the line pair $ax^2 + 2hxy + by^2 = 0$.

If the sum of the perpendicular distances from a variable point $P(x,y)$ to the two lines $x + y = 1$ and $x - y = 1$ is bounded at $2$ units, the locus track of $P$ forms a:

Find the value of the parameter $h$ if the general second-degree equation $x^2 + 2hxy + y^2 + 2x + 2y + 1 = 0$ factors into a pair of parallel straight lines.

A line is drawn through $P(1, 2)$ making an angle of $\theta$ with the positive direction of the x-axis. If it intersects the line $x + y = 4$ at a distance of $r$ units from $P$, then $r$ matches:

The product of the perpendicular distances dropped from the origin $(0, 0)$ to the pair of lines represented by $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ is:

If the lines $x + 3y = 4$, $3x - y = 2$, and $kx + y = 0$ form a right-angled triangle where the line $kx+y=0$ contains the hypotenuse, find the value of the perpendicular slope parameter $k$ if it matches the first line's orthogonal direction.

Find the total number of points of intersection that a family of lines $L_1 + \lambda L_2 = 0$ has across all possible real values of $\lambda$.

If the joint equation $x^2 - 2xy - y^2 = 0$ represents the angle bisectors of the line pair $ax^2 + 2xy + by^2 = 0$, find the value of the sum of coefficients $a + b$.

Assertion (A): The symmetric form of a straight line $\frac{x - x_1}{\cos\theta} = \frac{y - y_1}{\sin\theta} = r$ tracks geometric displacement, where $|r|$ represents the absolute distance between $(x,y)$ and $(x_1, y_1)$.
Reason (R): Squaring and adding the parametric shifts confirms that $(x-x_1)^2 + (y-y_1)^2 = r^2(\cos^2\theta + \sin^2\theta) = r^2$.

Assertion (A): The general second-degree equation represents a pair of straight lines if its matrix determinant discriminant satisfies $\Delta = 0$.
Reason (R): A vanishing determinant ensures that the quadratic expression can be factored cleanly into two real or imaginary linear polynomial equations.