Straight lines practice · Mixed difficulty

One possible condition for the three points $(a, b),(b, a)$ and $\left(a^2,-b^2\right)$ to be collinear is

If the lines $ax + 2\;y + 1 = 0,\;bx + 3\;y + 1 = 0,\;cx + 4\;y + 1 = 0$ are concurrent, then a,b,c are in

The point on the line $3x + 4y = 5$, which is equidistant from (1, 2) and (3, 4) is

If the roots of the equation $a x^2+b x+10=0$ are not real and distinct where $a, b \in R$ and $\alpha$ and $\beta$ are the values of $a$ and $b$ for which $5 a+b$ is minimum, then the family of lines $\alpha(4 x+2 y+3)+\beta(x-y-1)=0$ are concurrent at

If the points $\left( {1,1} \right),\left( { - 1, - 1} \right),\left( { - \sqrt 3 ,\sqrt 3 } \right)$ are the vertices of a triangle, then this triangle is

The points $(1,1),(-2,7)$ and $(3,-3)$

The equation of the image of the lines $y = \left| x \right|$ in the line mirror $x = 2$ is

If the straight line $ax + by + c = 0$ always passes through (1,-2), then a,b,c are

The equation of line passing through $\left(-1, \frac{\pi}{2}\right)$ and perpendicular to $\sqrt{3} \sin \theta+2 \cos \theta=\frac{4}{r}$ is

If the sum of distances of a point from two perpendicular lines is unity, then its locus is

The points $P(a, b), Q(c, d), R(a, d)$ and $S(c, b),$ where $a, b, c, d$ are distinct real numbers, are

The equation of the line passing through the intersection of $x - \sqrt 3 \;y + \sqrt 3 - 1 = 0$ and $x + y - 2 = 0$ and making an angle of ${15^ \circ }$ with the first line is

The distance between the parallel lines $3 x-4 y+7=0$ and $3 {x}-4 {y}+5=0$ is $\frac{{a}}{{b}}$. Value of ${a}+{b}$ is

If $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ represents a pair of straight lines equidistant from the origin, then

If $a > 0,\;b > 0$ the maximum area of the triangle formed by the points $O\left( {0,\;0} \right),\;A\left( {a\cos \theta ,\;b\sin \theta } \right)$ and $B\left( {a\cos \theta ,\; - b\sin \theta } \right)$ is (in sq unit)

If $P = \left( {1,0} \right),Q = \left( { - 1,0} \right)$ and $R = \left( {2,0} \right)$ are three given points, then the locus of the point $S\left( {x,y} \right)$ satisfying the relation $S{Q^2} + + S{R^2} = 2S{P^2}$ is

The line L given by $\frac{x}{5} + \frac{y}{b} = 1$ passes through the point (13, 32). The line K is parallel to L and has the equation $\frac{x}{c} + \frac{y}{3} - 1$. Then, the distance between L and K is

The line $3x + 2y = 24$ meets y-axis at A and x-axis at B. The perpendicular bisector of AB meets the line through (0,-1) parallel to x-axis at C. The area of the triangle ABC is

If area of triangle with vertices (0, 0), (0, 6) and $\left( {\mathrm{\alpha },\mathrm{\;\beta }} \right)$ is 15 sq unit, then

If a straight line L is perpendicular to the line $5x - y = 1$ such that the area of the $\Delta$ formed by the line L and the coordinate axes is 5, then the equation of the line L is

Consider the following statements.

The three given points $A, B, C$ are collinear i.e., lie on the same straight line, if

I. area of $\triangle A B C$ is zero.

II. slope of $A B=$ Slope of $B C$.

III. any one of the three points lie on the straight line joining the other two points.

Choose the correct option.

The equation ${x^3} - 6{x^2}y + 11x{y^2} - 6{y^3} = 0$ represents three straight lines passing through the origin, the slopes of which form an

The image of a point $A(3,8)$ in the line $x+3 y-7=0$, is $(a, b)$. Find $a-b$.

The area of triangle formed by the points $\left( {a,\;b + c} \right),\;\left( {b,\;c + a} \right),\;\left( {c,\;a + b} \right)$ is equal to

Equation of the straight line making equal intercepts on the axes and passing through the point (2, 4), is

Equation of the straight line cutting off an intercept 2 from the negative direction of the axis of y and inclined at ${30^ \circ }$ to the positive direction of x-axis, is

Consider the following statements.

I. The equation of a straight line passing through the point $\left(x_1, y_1\right)$ and having slope ${m}$ is given by $y-y_1=m\left(x-x_1\right)$

II. Equation of the $y$-axis is $x=0$.

Choose the correct option.

$A\left( { - 5,0} \right)$ and $B\left( {3,0} \right)$ are two of the vertices of a triangle ABC. Its area is 20 square cms. The vertex C lies on the line $x - y = 2.$. The coordinates of C are

A line passes through the point of intersection of the lines $3x + y + 1 = 0$ and $2x - y + 3 = 0$ and makes equal intercepts with axes. Then, equation of the line is

The point $(4,1)$ undergoes the following two successive transformation
(i) Reflection about the line $y=x$
(ii) Translation through a distance 2 units along the positive $x$-axis
Then the final coordinates of the point are $(x, y)$. Find $x+y$.