Straight Lines

By the given conditions, we can take two perpendicular lines as x and y axes. If (h, k) is any point on the locus, then Therefore, the locus is
This consist of a square of side 1.
Hence, the required locus is a square.

Orthocentre of right angled triangle is at the vertex of right angle. Therefore, orthocentre of the triangle is at (0 , 0 ).

PQRS is a parallelogram if and only if the mid point of the diagonals PR is same as that of the mid-point of QS.
That is, if and only if

Let S be the mid-point of QR and given PQR is an isosceles.
Therefore, PS QR and S is mid-point of hypotenuse,
S is equidistant from P, Q, R.

SInce,
and
But


Now, slope of QR is - 2 .
But
Slope of PS is 1/2.
Let m be the slope of P



Equations of PQ and PR are and
or 3 ( y - 1) + (x - 2) = 0
Therefore, joint equation of PQ and PR is
[ 3 (x - 2) - ( y - 1) ] [ (x - 2) + 3 ( y - 1) ] = 0

The correct answer is B:
Given that;
The equation of the line is:-
and
Let the equation of the line passes through the origin:
Then two intersection of (i) and (iii)

similarly, (ii) and (iii);

Let us consider a point ‘O’ denotes PQ in the ratio of
and

Let The equation of the tangent at P (1, 1) to the curve 2y = is Its meet the coordinate axes at and Area of OAB =

Given,


and x = - y + 1

From the graph, it is clear that equation of angle bisectors are and
Area of region bounded b y x + y = 3 ,x = 0 and y = 1 is
sq units.

Since, Again, AE = AD sec AE is HM of b and c EF = ED +D F = 2DE =2 AD tan = 2 Since, AD EF and DE = DF and AD is bisector. AEF is isosceles. Hence, (a), (b), (c), (d) are correct answers.

Since, triangle is isosceles, hence centroid is the desired point.
Coordinates of R

Let a and b be non-zero real numbers
Therefore the given equation
implies either



and x = 3y
represent two straight lines passing through origin or
when c = 0 and a and b are of same
signs, then

x = 0

and y =- 0
which is a point specified as the origin.
When, a = b and c is of sign opposite to that of a
represents a circle.
Hence, the given equation

may represent two straight lines and a circle.